{
  "meta": {
    "schema_version": 6,
    "title": "Clockwork/coloring correspondence",
    "source_catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/data/catalog.json",
    "source_catalog_sha256": "040eebe747815557014c1dbf1d4265d204aaae35c110595f2a15b94ee7f68ca0",
    "source_catalog_total_groups": 275,
    "selection": "group.forward == true",
    "forward_groups": 68,
    "traditional_color_classes_after_clock_inversion": 64,
    "definition": "kappa(M,v) = N*tau mod N; K = ker(kappa); regular action has H = K; ToS type is G for N=1, G/K for N=2, and G^N/K for N>2",
    "book_audit_counts": {
      "plane-group": 17,
      "direct-table": 39,
      "internal-discrepancy": 3,
      "composite-extension": 9
    },
    "book": {
      "title": "The Symmetries of Things",
      "authors": [
        "John H. Conway",
        "Heidi Burgiel",
        "Chaim Goodman-Strauss"
      ],
      "edition": 2008,
      "record_url": "https://books.google.com/books?id=EtQCk0TNafsC",
      "errata_url": "https://www.mit.edu/~hlb/Symmetries_of_Things/SoTerrors.html",
      "note": "Printed-page links open local highlighted evidence crops with Google Books as the no-JavaScript and complete-page fallback; attached-PDF indices are stored separately for audit reproducibility.",
      "annotated_excerpt_count": 65
    },
    "kernel_method": "Classify the tau=0 spatial operations plus their own translation lattice against the 17 canonical wallpaper groups.",
    "image_palette": [
      "#0072B2",
      "#E69F00",
      "#009E73",
      "#CC79A7",
      "#D55E00",
      "#56B4E9"
    ],
    "image_size": [
      720,
      420
    ],
    "signature_evidence_counts": {
      "onefold": 17,
      "exact-printed": 26,
      "type-representative": 13,
      "book-internal-discrepancy": 3,
      "rule-extension": 9
    }
  },
  "groups": [
    {
      "ordinal": 1,
      "id": "g1",
      "symbol": "o",
      "system": "Triclinic",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "◦",
      "book_color_signature": "◦",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [],
      "clockwork_description": "The direct-product lift over plane orbifold ◦ has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group ◦, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists ◦ among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::◦"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g1",
      "image": "output/clockwork-colorings/g1.webp",
      "image_alt": "Static perfect 1-colouring for group g1: asymmetric motifs carry phase colours for Conway type ◦.",
      "render": {
        "basis": [
          [
            0.787524,
            0.212476
          ],
          [
            0.212476,
            0.787524
          ]
        ],
        "ops": [
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.0851,
          0.0785
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 1,
        "template": "omitted_trivial_time_action",
        "generators": [],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 2,
      "id": "g5",
      "symbol": "2222",
      "system": "T-Monoclinic",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "2222",
      "book_color_signature": "2222",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 2222 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 2222, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 2222 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::2222"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g5",
      "image": "output/clockwork-colorings/g5.webp",
      "image_alt": "Static perfect 1-colouring for group g5: asymmetric motifs carry phase colours for Conway type 2222.",
      "render": {
        "basis": [
          [
            2.416829,
            -1.416829
          ],
          [
            -1.416829,
            2.416829
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.847,
          0.1182
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 2,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 3,
      "id": "g6",
      "symbol": "2₁2₁2₁2₁",
      "system": "T-Monoclinic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "2222/◦",
      "book_color_signature": "²2²2²2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 2222 onto C2. The operation phases are 2-fold rotations: 1/2.",
      "coloring_description": "The book colour type is 2222/◦: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = ◦. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "2222/◦ appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::2222/◦"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g6",
      "image": "output/clockwork-colorings/g6.webp",
      "image_alt": "Static perfect 2-colouring for group g6: asymmetric motifs carry phase colours for Conway type 2222/◦.",
      "render": {
        "basis": [
          [
            2.416829,
            -1.416829
          ],
          [
            -1.416829,
            2.416829
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.847,
          0.1182
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 2,
        "template": "cyclic_2",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = 1"
      }
    },
    {
      "ordinal": 4,
      "id": "g7",
      "symbol": "c222₁2₁",
      "system": "T-Monoclinic",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": true,
      "parent": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "2222/2222",
      "book_color_signature": "¹2¹2²2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 2222 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2.",
      "coloring_description": "The book colour type is 2222/2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "2222/2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::2222/2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g7",
      "image": "output/clockwork-colorings/g7.webp",
      "image_alt": "Static perfect 2-colouring for group g7: asymmetric motifs carry phase colours for Conway type 2222/2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.228,
          0.8642
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 6 equivalent generator signatures under the same colour type 2222/2222. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 6
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "elementary_2_2",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "1/2-cell translation along b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = 1; AB = BA"
      }
    },
    {
      "ordinal": 5,
      "id": "g8",
      "symbol": "*×",
      "system": "R-Monoclinic",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "*×",
        "hm": "cm"
      },
      "kernel": {
        "orbifold": "*×",
        "hm": "cm"
      },
      "tos_notation": "*×",
      "book_color_signature": "*×",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold *× has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group *×, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists *× among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::*×"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g8",
      "image": "output/clockwork-colorings/g8.webp",
      "image_alt": "Static perfect 1-colouring for group g8: asymmetric motifs carry phase colours for Conway type *×.",
      "render": {
        "basis": [
          [
            0.707106781187,
            0.707106781187
          ],
          [
            -0.707106781187,
            0.707106781187
          ]
        ],
        "ops": [
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.1566,
          0.65
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 2,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 6,
      "id": "g9",
      "symbol": "~*×½",
      "system": "R-Monoclinic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*×",
        "hm": "cm"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "*×/◦",
      "book_color_signature": "*²×²",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *× onto C2. The operation phases are reflections / glides: 1/2.",
      "coloring_description": "The book colour type is *×/◦: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = ◦. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*×/◦ appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::*×/◦"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g9",
      "image": "output/clockwork-colorings/g9.webp",
      "image_alt": "Static perfect 2-colouring for group g9: asymmetric motifs carry phase colours for Conway type *×/◦.",
      "render": {
        "basis": [
          [
            0.707106781187,
            0.707106781187
          ],
          [
            -0.707106781187,
            0.707106781187
          ]
        ],
        "ops": [
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.1566,
          0.65
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 2,
        "template": "cyclic_2",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = 1"
      }
    },
    {
      "ordinal": 7,
      "id": "g10",
      "symbol": "**",
      "system": "R-Monoclinic",
      "bravais": "C-centred (spatial)",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "**",
        "hm": "pm"
      },
      "kernel": {
        "orbifold": "**",
        "hm": "pm"
      },
      "tos_notation": "**",
      "book_color_signature": "**",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold ** has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group **, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists ** among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::**"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g10",
      "image": "output/clockwork-colorings/g10.webp",
      "image_alt": "Static perfect 1-colouring for group g10: asymmetric motifs carry phase colours for Conway type **.",
      "render": {
        "basis": [
          [
            0.707107,
            0.346697
          ],
          [
            -0.707107,
            0.346697
          ]
        ],
        "ops": [
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.0851,
          0.8404
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 8,
      "id": "g11",
      "symbol": "××",
      "system": "R-Monoclinic",
      "bravais": "C-centred (spatial)",
      "product": true,
      "symmorphic": false,
      "parent": {
        "orbifold": "××",
        "hm": "pg"
      },
      "kernel": {
        "orbifold": "××",
        "hm": "pg"
      },
      "tos_notation": "××",
      "book_color_signature": "××",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold ×× has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group ××, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists ×× among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::××"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g11",
      "image": "output/clockwork-colorings/g11.webp",
      "image_alt": "Static perfect 1-colouring for group g11: asymmetric motifs carry phase colours for Conway type ××.",
      "render": {
        "basis": [
          [
            0.707107,
            1.228417
          ],
          [
            -0.707107,
            1.228417
          ]
        ],
        "ops": [
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.25,
              0.25
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.75,
              0.75
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.0851,
          0.8404
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "glide",
            "operation": "Glide reflection",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 9,
      "id": "g54",
      "symbol": "*2222",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "kernel": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "tos_notation": "*2222",
      "book_color_signature": "*2222",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold *2222 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group *2222, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists *2222 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::*2222"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g54",
      "image": "output/clockwork-colorings/g54.webp",
      "image_alt": "Static perfect 1-colouring for group g54: asymmetric motifs carry phase colours for Conway type *2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 10,
      "id": "g55",
      "symbol": "*~2~2~2~2",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "*2222/2222",
      "book_color_signature": "*²2²2²2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *2222 onto C2. The operation phases are 2-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is *2222/2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*2222/2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*2222/2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g55",
      "image": "output/clockwork-colorings/g55.webp",
      "image_alt": "Static perfect 2-colouring for group g55: asymmetric motifs carry phase colours for Conway type *2222/2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "elementary_2_2",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = 1; AB = BA"
      }
    },
    {
      "ordinal": 11,
      "id": "g56",
      "symbol": "22*",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": true,
      "symmorphic": false,
      "parent": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "kernel": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "tos_notation": "22*",
      "book_color_signature": "22*",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 22* has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 22*, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 22* among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::22*"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g56",
      "image": "output/clockwork-colorings/g56.webp",
      "image_alt": "Static perfect 1-colouring for group g56: asymmetric motifs carry phase colours for Conway type 22*.",
      "render": {
        "basis": [
          [
            0.0,
            -1.0
          ],
          [
            1.0,
            0.0
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 12,
      "id": "g57",
      "symbol": "22~*",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "22*/2222",
      "book_color_signature": "¹2¹2*²",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 22* onto C2. The operation phases are 2-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is 22*/2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "22*/2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::22*/2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g57",
      "image": "output/clockwork-colorings/g57.webp",
      "image_alt": "Static perfect 2-colouring for group g57: asymmetric motifs carry phase colours for Conway type 22*/2222.",
      "render": {
        "basis": [
          [
            0.0,
            -1.0
          ],
          [
            1.0,
            0.0
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "elementary_2_2",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = 1; AB = BA"
      }
    },
    {
      "ordinal": 13,
      "id": "g58",
      "symbol": "22×",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": true,
      "symmorphic": false,
      "parent": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "kernel": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "tos_notation": "22×",
      "book_color_signature": "22×",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 22× has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 22×, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 22× among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::22×"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g58",
      "image": "output/clockwork-colorings/g58.webp",
      "image_alt": "Static perfect 1-colouring for group g58: asymmetric motifs carry phase colours for Conway type 22×.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.3232,
          0.4357
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "glide",
            "operation": "Glide reflection",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 14,
      "id": "g59",
      "symbol": "22×½",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "22×/2222",
      "book_color_signature": "¹2¹2ײ",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 22× onto C2. The operation phases are 2-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is 22×/2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "22×/2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::22×/2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g59",
      "image": "output/clockwork-colorings/g59.webp",
      "image_alt": "Static perfect 2-colouring for group g59: asymmetric motifs carry phase colours for Conway type 22×/2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.3232,
          0.4357
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "elementary_2_2",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "glide",
            "operation": "Glide reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = 1; AB = BA"
      }
    },
    {
      "ordinal": 15,
      "id": "g60",
      "symbol": "*2₁~2₁2₁~2₁",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "kernel": {
        "orbifold": "**",
        "hm": "pm"
      },
      "tos_notation": "*2222/**",
      "book_color_signature": "*¹2²2¹2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *2222 onto C2. The operation phases are 2-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *2222/**: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = **. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*2222/** appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*2222/**"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g60",
      "image": "output/clockwork-colorings/g60.webp",
      "image_alt": "Static perfect 2-colouring for group g60: asymmetric motifs carry phase colours for Conway type *2222/**.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type *2222/**. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "elementary_2_2",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = 1; AB = BA"
      }
    },
    {
      "ordinal": 16,
      "id": "g61",
      "symbol": "2₁2₁~*",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "kernel": {
        "orbifold": "××",
        "hm": "pg"
      },
      "tos_notation": "22*/××",
      "book_color_signature": "²2²2*²",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 22* onto C2. The operation phases are 2-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 22*/××: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = ××. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "22*/×× appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::22*/××"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g61",
      "image": "output/clockwork-colorings/g61.webp",
      "image_alt": "Static perfect 2-colouring for group g61: asymmetric motifs carry phase colours for Conway type 22*/××.",
      "render": {
        "basis": [
          [
            0.0,
            -1.0
          ],
          [
            1.0,
            0.0
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "elementary_2_2",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = 1; AB = BA"
      }
    },
    {
      "ordinal": 17,
      "id": "g62",
      "symbol": "2₁2₁*",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "kernel": {
        "orbifold": "**",
        "hm": "pm"
      },
      "tos_notation": "22*/**",
      "book_color_signature": "²2²2*¹",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 22* onto C2. The operation phases are 2-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 22*/**: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = **. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "22*/** appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::22*/**"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g62",
      "image": "output/clockwork-colorings/g62.webp",
      "image_alt": "Static perfect 2-colouring for group g62: asymmetric motifs carry phase colours for Conway type 22*/**.",
      "render": {
        "basis": [
          [
            0.0,
            -1.0
          ],
          [
            1.0,
            0.0
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "elementary_2_2",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = 1; AB = BA"
      }
    },
    {
      "ordinal": 18,
      "id": "g63",
      "symbol": "2₁2₁×",
      "system": "Orthorhombic",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "kernel": {
        "orbifold": "××",
        "hm": "pg"
      },
      "tos_notation": "22×/××",
      "book_color_signature": "²2²2×¹",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 22× onto C2. The operation phases are 2-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 22×/××: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = ××. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "22×/×× appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::22×/××"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g63",
      "image": "output/clockwork-colorings/g63.webp",
      "image_alt": "Static perfect 2-colouring for group g63: asymmetric motifs carry phase colours for Conway type 22×/××.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.3232,
          0.4357
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type 22×/××. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "elementary_2_2",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "glide",
            "operation": "Glide reflection",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = 1; AB = BA"
      }
    },
    {
      "ordinal": 19,
      "id": "g64",
      "symbol": "c*222₁2₁ᵃ",
      "system": "Orthorhombic",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": true,
      "parent": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "kernel": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "tos_notation": "*2222/*2222",
      "book_color_signature": "*¹2¹2¹2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *2222 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *2222/*2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = *2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*2222/*2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*2222/*2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g64",
      "image": "output/clockwork-colorings/g64.webp",
      "image_alt": "Static perfect 2-colouring for group g64: asymmetric motifs carry phase colours for Conway type *2222/*2222.",
      "render": {
        "basis": [
          [
            1.0,
            0.0
          ],
          [
            0.0,
            2.204541
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.7518,
          0.3642
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 4 equivalent generator signatures under the same colour type *2222/*2222. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 4
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "1/2-cell translation along b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 20,
      "id": "g65",
      "symbol": "c*222₁2₁ᵇ",
      "system": "Orthorhombic",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "kernel": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "tos_notation": "*2222/22*",
      "book_color_signature": "*¹2²2²2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *2222 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *2222/22*: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 22*. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*2222/22* appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*2222/22*"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g65",
      "image": "output/clockwork-colorings/g65.webp",
      "image_alt": "Static perfect 2-colouring for group g65: asymmetric motifs carry phase colours for Conway type *2222/22*.",
      "render": {
        "basis": [
          [
            0.0,
            -1.0
          ],
          [
            2.204541,
            -0.0
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.7518,
          0.3642
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 4 equivalent generator signatures under the same colour type *2222/22*. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 4
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "1/2-cell translation along b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 21,
      "id": "g66",
      "symbol": "c22₁*ᵃ",
      "system": "Orthorhombic",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "kernel": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "tos_notation": "22*/22*",
      "book_color_signature": "¹2²2*¹",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 22* onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 22*/22*: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 22*. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "22*/22* appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::22*/22*"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g66",
      "image": "output/clockwork-colorings/g66.webp",
      "image_alt": "Static perfect 2-colouring for group g66: asymmetric motifs carry phase colours for Conway type 22*/22*.",
      "render": {
        "basis": [
          [
            1.0,
            0.0
          ],
          [
            0.0,
            0.498063
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.3708,
          0.65
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type 22*/22*. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "1/2-cell translation along b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "C",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 22,
      "id": "g67",
      "symbol": "c22₁*ᵇ",
      "system": "Orthorhombic",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "kernel": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "tos_notation": "22*/22×",
      "book_color_signature": "¹2²2*²",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 22* onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 22*/22×: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 22×. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "22*/22× appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::22*/22×"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g67",
      "image": "output/clockwork-colorings/g67.webp",
      "image_alt": "Static perfect 2-colouring for group g67: asymmetric motifs carry phase colours for Conway type 22*/22×.",
      "render": {
        "basis": [
          [
            1.0,
            0.0
          ],
          [
            0.0,
            0.498063
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.3708,
          0.65
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type 22*/22×. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "1/2-cell translation along b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "C",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 23,
      "id": "g68",
      "symbol": "2*22",
      "system": "Orthorhombic",
      "bravais": "C-centred (spatial)",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "kernel": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "tos_notation": "2*22",
      "book_color_signature": "2*22",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 2*22 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 2*22, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 2*22 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::2*22"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g68",
      "image": "output/clockwork-colorings/g68.webp",
      "image_alt": "Static perfect 1-colouring for group g68: asymmetric motifs carry phase colours for Conway type 2*22.",
      "render": {
        "basis": [
          [
            1.0,
            0.0
          ],
          [
            0.0,
            2.204541
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.7518,
          0.3642
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 24,
      "id": "g69",
      "symbol": "2*~2~2",
      "system": "Orthorhombic",
      "bravais": "C-centred (spatial)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "2*22/2222",
      "book_color_signature": "¹2*²2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 2*22 onto C2. The operation phases are translations: 0; 2-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is 2*22/2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "2*22/2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::2*22/2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g69",
      "image": "output/clockwork-colorings/g69.webp",
      "image_alt": "Static perfect 2-colouring for group g69: asymmetric motifs carry phase colours for Conway type 2*22/2222.",
      "render": {
        "basis": [
          [
            1.0,
            0.0
          ],
          [
            0.0,
            2.204541
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.7518,
          0.3642
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 25,
      "id": "g70",
      "symbol": "2₁*2₁~2₁",
      "system": "Orthorhombic",
      "bravais": "C-centred (spatial)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "kernel": {
        "orbifold": "*×",
        "hm": "cm"
      },
      "tos_notation": "2*22/*×",
      "book_color_signature": "²2*¹2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 2*22 onto C2. The operation phases are translations: 0; 2-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 2*22/*×: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = *×. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "2*22/*× appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::2*22/*×"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g70",
      "image": "output/clockwork-colorings/g70.webp",
      "image_alt": "Static perfect 2-colouring for group g70: asymmetric motifs carry phase colours for Conway type 2*22/*×.",
      "render": {
        "basis": [
          [
            1.0,
            0.0
          ],
          [
            0.0,
            2.204541
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.7518,
          0.3642
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type 2*22/*×. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 26,
      "id": "g71",
      "symbol": "c2₁*22ᵃ",
      "system": "Orthorhombic",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": true,
      "parent": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "kernel": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "tos_notation": "2*22/*2222",
      "book_color_signature": "²2*¹2¹2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 2*22 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 2*22/*2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = *2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "2*22/*2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::2*22/*2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g71",
      "image": "output/clockwork-colorings/g71.webp",
      "image_alt": "Static perfect 2-colouring for group g71: asymmetric motifs carry phase colours for Conway type 2*22/*2222.",
      "render": {
        "basis": [
          [
            1.0,
            0.0
          ],
          [
            0.0,
            2.204541
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.7518,
          0.3642
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 27,
      "id": "g72",
      "symbol": "c2₁*22ᵇ",
      "system": "Orthorhombic",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "kernel": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "tos_notation": "2*22/22×",
      "book_color_signature": "²2*²2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 2*22 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 2*22/22×: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 22×. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "2*22/22× appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::2*22/22×"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g72",
      "image": "output/clockwork-colorings/g72.webp",
      "image_alt": "Static perfect 2-colouring for group g72: asymmetric motifs carry phase colours for Conway type 2*22/22×.",
      "render": {
        "basis": [
          [
            1.0,
            0.0
          ],
          [
            0.0,
            2.204541
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.7518,
          0.3642
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 28,
      "id": "g73",
      "symbol": "c2*2₁2₁",
      "system": "Orthorhombic",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "kernel": {
        "orbifold": "22*",
        "hm": "pmg"
      },
      "tos_notation": "2*22/22*",
      "book_color_signature": "¹2*¹2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 2*22 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 2*22/22*: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 22*. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "2*22/22* appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::2*22/22*"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g73",
      "image": "output/clockwork-colorings/g73.webp",
      "image_alt": "Static perfect 2-colouring for group g73: asymmetric motifs carry phase colours for Conway type 2*22/22*.",
      "render": {
        "basis": [
          [
            -0.0,
            -1.0
          ],
          [
            0.498063,
            0.0
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.3708,
          0.5071
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type 2*22/22*. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "elementary_2_3",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = C² = 1; AB = BA, AC = CA, BC = CB"
      }
    },
    {
      "ordinal": 29,
      "id": "g74",
      "symbol": "c*22₁22₁",
      "system": "Orthorhombic",
      "bravais": "F-centred",
      "product": false,
      "symmorphic": true,
      "parent": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "kernel": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "tos_notation": "*2222/2*22",
      "book_color_signature": "*¹2¹2²2²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *2222 onto C2. The operation phases are translations: 0, 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *2222/2*22: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2*22. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*2222/2*22 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*2222/2*22"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g74",
      "image": "output/clockwork-colorings/g74.webp",
      "image_alt": "Static perfect 2-colouring for group g74: asymmetric motifs carry phase colours for Conway type *2222/2*22.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.3708,
          0.3642
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 4 equivalent generator signatures under the same colour type *2222/2*22. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 4
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 16,
        "template": "elementary_2_4",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "1/2-cell translation along b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "translation",
            "operation": "1/2-cell translation along a",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "D",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = C² = D² = 1; AB = BA, AC = CA, AD = DA, BC = CB, BD = DB, CD = DC"
      }
    },
    {
      "ordinal": 30,
      "id": "g75",
      "symbol": "c22₁×¼",
      "system": "Orthorhombic",
      "bravais": "F-centred",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "22×⁴/2222",
      "book_color_signature": "¹2²2×⁴",
      "clock_order": 4,
      "cyclic_group": "C_4",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/4",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/2",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "3/4",
          "color": "#CC79A7"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/4",
            "3/4"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 22× onto C4. The operation phases are translations: 0, 1/2; 2-fold rotations: 0, 1/2; reflections / glides: 1/4, 3/4.",
      "coloring_description": "The book-style colour type is 22×⁴/2222: the exponent 4 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 4 and G/K is isomorphic to C4. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/4 sends colour k to k+j modulo 4.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C4 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 22×⁴/2222. Its prime-index layers 22×/2222 then 2222/2222 are tabulated; the checked phase image establishes that their extension is cyclic C4. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "22×/2222",
            "index": 2,
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::22×/2222"
          },
          {
            "notation": "2222/2222",
            "index": 2,
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::2222/2222"
          }
        ],
        "intermediate_orbifold": "2222"
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g75",
      "image": "output/clockwork-colorings/g75.webp",
      "image_alt": "Static perfect 4-colouring for group g75: asymmetric motifs carry phase colours for Conway type 22×⁴/2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.25,
              0.25
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.25,
              0.75
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.75,
              0.25
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.75,
              0.75
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.25
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.75
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.25,
              0.0
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.25,
              0.5
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.75,
              0.0
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.75,
              0.5
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2756,
          0.2214
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C4 extension",
        "summary": "The book does not enumerate composite C4 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 16,
        "template": "exceptional_16",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "Half-turn rotation (180°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "glide",
            "operation": "Glide reflection",
            "phase": "3/4",
            "time_shift": "+3/4 period"
          }
        ],
        "relations": "A² = B⁴ = (AB)⁴ = 1; AB² = B²A"
      }
    },
    {
      "ordinal": 31,
      "id": "g94",
      "symbol": "442",
      "system": "Tetragonal",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "442",
        "hm": "p4"
      },
      "kernel": {
        "orbifold": "442",
        "hm": "p4"
      },
      "tos_notation": "442",
      "book_color_signature": "442",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 442 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 442, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 442 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::442"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g94",
      "image": "output/clockwork-colorings/g94.webp",
      "image_alt": "Static perfect 1-colouring for group g94: asymmetric motifs carry phase colours for Conway type 442.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.3232,
          0.8166
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 32,
      "id": "g95",
      "symbol": "4₂4₂2",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "442",
        "hm": "p4"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "442/2222",
      "book_color_signature": "²4²4¹2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 442 onto C2. The operation phases are 2-fold rotations: 0; 4-fold rotations: 1/2.",
      "coloring_description": "The book colour type is 442/2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "442/2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::442/2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g95",
      "image": "output/clockwork-colorings/g95.webp",
      "image_alt": "Static perfect 2-colouring for group g95: asymmetric motifs carry phase colours for Conway type 442/2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.3232,
          0.8166
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "cyclic_4",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A⁴ = 1"
      }
    },
    {
      "ordinal": 33,
      "id": "g96",
      "symbol": "4₁4₁2₁",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "442",
        "hm": "p4"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "442⁴/◦",
      "book_color_signature": "⁴4⁴4²2",
      "clock_order": 4,
      "cyclic_group": "C_4",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/4",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/2",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "3/4",
          "color": "#CC79A7"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/4",
            "3/4"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 442 onto C4. The operation phases are 2-fold rotations: 1/2; 4-fold rotations: 1/4, 3/4.",
      "coloring_description": "The book-style colour type is 442⁴/◦: the exponent 4 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = ◦. Here [G:K] = 4 and G/K is isomorphic to C4. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/4 sends colour k to k+j modulo 4.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C4 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 442⁴/◦. Its prime-index layers 442/2222 then 2222/◦ are tabulated; the checked phase image establishes that their extension is cyclic C4. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "442/2222",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::442/2222"
          },
          {
            "notation": "2222/◦",
            "index": 2,
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::2222/◦"
          }
        ],
        "intermediate_orbifold": "2222"
      },
      "inverse_clock_mate": "g97",
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g96",
      "image": "output/clockwork-colorings/g96.webp",
      "image_alt": "Static perfect 4-colouring for group g96: asymmetric motifs carry phase colours for Conway type 442⁴/◦.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.3232,
          0.8166
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C4 extension",
        "summary": "The book does not enumerate composite C4 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "cyclic_4",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "1/4",
            "time_shift": "+1/4 period"
          }
        ],
        "relations": "A⁴ = 1"
      }
    },
    {
      "ordinal": 34,
      "id": "g97",
      "symbol": "4₃4₃2₁",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "442",
        "hm": "p4"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "442⁴/◦",
      "book_color_signature": "⁴4⁴4²2",
      "clock_order": 4,
      "cyclic_group": "C_4",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/4",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/2",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "3/4",
          "color": "#CC79A7"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/4",
            "3/4"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 442 onto C4. The operation phases are 2-fold rotations: 1/2; 4-fold rotations: 1/4, 3/4.",
      "coloring_description": "The book-style colour type is 442⁴/◦: the exponent 4 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = ◦. Here [G:K] = 4 and G/K is isomorphic to C4. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/4 sends colour k to k+j modulo 4.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C4 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 442⁴/◦. Its prime-index layers 442/2222 then 2222/◦ are tabulated; the checked phase image establishes that their extension is cyclic C4. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "442/2222",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::442/2222"
          },
          {
            "notation": "2222/◦",
            "index": 2,
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::2222/◦"
          }
        ],
        "intermediate_orbifold": "2222"
      },
      "inverse_clock_mate": "g96",
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g97",
      "image": "output/clockwork-colorings/g97.webp",
      "image_alt": "Static perfect 4-colouring for group g97: asymmetric motifs carry phase colours for Conway type 442⁴/◦.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.3232,
          0.8166
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C4 extension",
        "summary": "The book does not enumerate composite C4 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 4,
        "template": "cyclic_4",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "3/4",
            "time_shift": "+3/4 period"
          }
        ],
        "relations": "A⁴ = 1"
      }
    },
    {
      "ordinal": 35,
      "id": "g98",
      "symbol": "c44₂2₁",
      "system": "Tetragonal",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": true,
      "parent": {
        "orbifold": "442",
        "hm": "p4"
      },
      "kernel": {
        "orbifold": "442",
        "hm": "p4"
      },
      "tos_notation": "442/442",
      "book_color_signature": "¹4²4²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 442 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; 4-fold rotations: 0, 1/2.",
      "coloring_description": "The book colour type is 442/442: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 442. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "442/442 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::442/442"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g98",
      "image": "output/clockwork-colorings/g98.webp",
      "image_alt": "Static perfect 2-colouring for group g98: asymmetric motifs carry phase colours for Conway type 442/442.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.7518,
          0.0309
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type 442/442. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "cyclic_2_x_4",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B⁴ = 1; AB = BA"
      }
    },
    {
      "ordinal": 36,
      "id": "g99",
      "symbol": "c4₁4₃2",
      "system": "Tetragonal",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "442",
        "hm": "p4"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "442⁴/2222",
      "book_color_signature": "⁴4⁴4¹2",
      "clock_order": 4,
      "cyclic_group": "C_4",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/4",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/2",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "3/4",
          "color": "#CC79A7"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/4",
            "3/4"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 442 onto C4. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; 4-fold rotations: 1/4, 3/4.",
      "coloring_description": "The book-style colour type is 442⁴/2222: the exponent 4 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 4 and G/K is isomorphic to C4. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/4 sends colour k to k+j modulo 4.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C4 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 442⁴/2222. Its prime-index layers 442/2222 then 2222/2222 are tabulated; the checked phase image establishes that their extension is cyclic C4. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "442/2222",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::442/2222"
          },
          {
            "notation": "2222/2222",
            "index": 2,
            "printed_page": 141,
            "pdf_page": 160,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA141",
            "excerpt_key": "p141::2222/2222"
          }
        ],
        "intermediate_orbifold": "2222"
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g99",
      "image": "output/clockwork-colorings/g99.webp",
      "image_alt": "Static perfect 4-colouring for group g99: asymmetric motifs carry phase colours for Conway type 442⁴/2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.25,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.75,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.25,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.75,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.4899,
          0.4357
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C4 extension",
        "summary": "The book does not enumerate composite C4 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "cyclic_2_x_4",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "3/4",
            "time_shift": "+3/4 period"
          }
        ],
        "relations": "A² = B⁴ = 1; AB = BA"
      }
    },
    {
      "ordinal": 37,
      "id": "g128",
      "symbol": "*442",
      "system": "Tetragonal",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "*442",
        "hm": "p4m"
      },
      "kernel": {
        "orbifold": "*442",
        "hm": "p4m"
      },
      "tos_notation": "*442",
      "book_color_signature": "*442",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold *442 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group *442, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists *442 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::*442"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g128",
      "image": "output/clockwork-colorings/g128.webp",
      "image_alt": "Static perfect 1-colouring for group g128: asymmetric motifs carry phase colours for Conway type *442.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.1566,
          0.65
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 38,
      "id": "g129",
      "symbol": "*4₂~4₂2",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*442",
        "hm": "p4m"
      },
      "kernel": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "tos_notation": "*442/*2222",
      "book_color_signature": "*¹4²4¹2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *442 onto C2. The operation phases are 2-fold rotations: 0; 4-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *442/*2222: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = *2222. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*442/*2222 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*442/*2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g129",
      "image": "output/clockwork-colorings/g129.webp",
      "image_alt": "Static perfect 2-colouring for group g129: asymmetric motifs carry phase colours for Conway type *442/*2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.1566,
          0.65
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "dihedral_4_reflections",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = (AB)⁴ = 1"
      }
    },
    {
      "ordinal": 39,
      "id": "g130",
      "symbol": "*~4₂4₂~2",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*442",
        "hm": "p4m"
      },
      "kernel": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "tos_notation": "*442/2*22",
      "book_color_signature": "*²4¹4²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *442 onto C2. The operation phases are 2-fold rotations: 0; 4-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *442/2*22: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2*22. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*442/2*22 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*442/2*22"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g130",
      "image": "output/clockwork-colorings/g130.webp",
      "image_alt": "Static perfect 2-colouring for group g130: asymmetric motifs carry phase colours for Conway type *442/2*22.",
      "render": {
        "basis": [
          [
            1.000000000001,
            -0.0
          ],
          [
            0.0,
            1.000000000001
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.1566,
          0.65
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "dihedral_4_reflections",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = (AB)⁴ = 1"
      }
    },
    {
      "ordinal": 40,
      "id": "g131",
      "symbol": "*~4~4~2",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*442",
        "hm": "p4m"
      },
      "kernel": {
        "orbifold": "442",
        "hm": "p4"
      },
      "tos_notation": "*442/442",
      "book_color_signature": "*²4²4²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *442 onto C2. The operation phases are 2-fold rotations: 0; 4-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is *442/442: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 442. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*442/442 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*442/442"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g131",
      "image": "output/clockwork-colorings/g131.webp",
      "image_alt": "Static perfect 2-colouring for group g131: asymmetric motifs carry phase colours for Conway type *442/442.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.1566,
          0.65
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "dihedral_4_reflections",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = (AB)⁴ = 1"
      }
    },
    {
      "ordinal": 41,
      "id": "g132",
      "symbol": "4*2",
      "system": "Tetragonal",
      "bravais": "P",
      "product": true,
      "symmorphic": false,
      "parent": {
        "orbifold": "4*2",
        "hm": "p4g"
      },
      "kernel": {
        "orbifold": "4*2",
        "hm": "p4g"
      },
      "tos_notation": "4*2",
      "book_color_signature": "4*2",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 4*2 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 4*2, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 4*2 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::4*2"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g132",
      "image": "output/clockwork-colorings/g132.webp",
      "image_alt": "Static perfect 1-colouring for group g132: asymmetric motifs carry phase colours for Conway type 4*2.",
      "render": {
        "basis": [
          [
            1.000000000001,
            -0.0
          ],
          [
            0.0,
            1.000000000001
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.5071
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 42,
      "id": "g133",
      "symbol": "4₂*~2",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "4*2",
        "hm": "p4g"
      },
      "kernel": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "tos_notation": "4*2/22×",
      "book_color_signature": "²4*²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 4*2 onto C2. The operation phases are 2-fold rotations: 0; 4-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 4*2/22×: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 22×. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "4*2/22× appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::4*2/22×"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g133",
      "image": "output/clockwork-colorings/g133.webp",
      "image_alt": "Static perfect 2-colouring for group g133: asymmetric motifs carry phase colours for Conway type 4*2/22×.",
      "render": {
        "basis": [
          [
            1.000000000001,
            -0.0
          ],
          [
            0.0,
            1.000000000001
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.5071
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "dihedral_4_rotation",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B⁴ = (AB)² = 1"
      }
    },
    {
      "ordinal": 43,
      "id": "g134",
      "symbol": "4₂*2",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "4*2",
        "hm": "p4g"
      },
      "kernel": {
        "orbifold": "2*22",
        "hm": "cmm"
      },
      "tos_notation": "4*2/2*22",
      "book_color_signature": "²4*¹2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 4*2 onto C2. The operation phases are 2-fold rotations: 0; 4-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is 4*2/2*22: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 2*22. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "4*2/2*22 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::4*2/2*22"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g134",
      "image": "output/clockwork-colorings/g134.webp",
      "image_alt": "Static perfect 2-colouring for group g134: asymmetric motifs carry phase colours for Conway type 4*2/2*22.",
      "render": {
        "basis": [
          [
            1.000000000001,
            -0.0
          ],
          [
            0.0,
            1.000000000001
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.5071
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "dihedral_4_rotation",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B⁴ = (AB)² = 1"
      }
    },
    {
      "ordinal": 44,
      "id": "g135",
      "symbol": "4*~2",
      "system": "Tetragonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "4*2",
        "hm": "p4g"
      },
      "kernel": {
        "orbifold": "442",
        "hm": "p4"
      },
      "tos_notation": "4*2/442",
      "book_color_signature": "¹4*²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 4*2 onto C2. The operation phases are 2-fold rotations: 0; 4-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is 4*2/442: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 442. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "4*2/442 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::4*2/442"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g135",
      "image": "output/clockwork-colorings/g135.webp",
      "image_alt": "Static perfect 2-colouring for group g135: asymmetric motifs carry phase colours for Conway type 4*2/442.",
      "render": {
        "basis": [
          [
            1.000000000001,
            -0.0
          ],
          [
            0.0,
            1.000000000001
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.2518,
          0.5071
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 8,
        "template": "dihedral_4_rotation",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B⁴ = (AB)² = 1"
      }
    },
    {
      "ordinal": 45,
      "id": "g136",
      "symbol": "c*44₂2₁ᵃ",
      "system": "Tetragonal",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": true,
      "parent": {
        "orbifold": "*442",
        "hm": "p4m"
      },
      "kernel": {
        "orbifold": "*442",
        "hm": "p4m"
      },
      "tos_notation": "*442/*442",
      "book_color_signature": "*¹4¹4²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *442 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; 4-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *442/*442: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = *442. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*442/*442 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*442/*442"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g136",
      "image": "output/clockwork-colorings/g136.webp",
      "image_alt": "Static perfect 2-colouring for group g136: asymmetric motifs carry phase colours for Conway type *442/*442.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.1089,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type *442/*442. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 16,
        "template": "cyclic_2_x_dihedral_4",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = C² = (BC)⁴ = 1; AB = BA, AC = CA"
      }
    },
    {
      "ordinal": 46,
      "id": "g137",
      "symbol": "c4₃*2",
      "system": "Tetragonal",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "4*2",
        "hm": "p4g"
      },
      "kernel": {
        "orbifold": "*2222",
        "hm": "pmm"
      },
      "tos_notation": "4*2⁴/*2222",
      "book_color_signature": "⁴4*¹2",
      "clock_order": 4,
      "cyclic_group": "C_4",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/4",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/2",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "3/4",
          "color": "#CC79A7"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/4",
            "3/4"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/4",
            "1/2",
            "3/4"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 4*2 onto C4. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; 4-fold rotations: 1/4, 3/4; reflections / glides: 0, 1/4, 1/2, 3/4.",
      "coloring_description": "The book-style colour type is 4*2⁴/*2222: the exponent 4 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = *2222. Here [G:K] = 4 and G/K is isomorphic to C4. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/4 sends colour k to k+j modulo 4.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C4 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 4*2⁴/*2222. Its prime-index layers 4*2/2*22 then 2*22/*2222 are tabulated; the checked phase image establishes that their extension is cyclic C4. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "4*2/2*22",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::4*2/2*22"
          },
          {
            "notation": "2*22/*2222",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::2*22/*2222"
          }
        ],
        "intermediate_orbifold": "2*22"
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g137",
      "image": "output/clockwork-colorings/g137.webp",
      "image_alt": "Static perfect 4-colouring for group g137: asymmetric motifs carry phase colours for Conway type 4*2⁴/*2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.25,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.75,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.25,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.75,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.25,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.75,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.25,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.75,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.1327,
          0.3881
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C4 extension",
        "summary": "The book does not enumerate composite C4 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 16,
        "template": "exceptional_16",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "3/4",
            "time_shift": "+3/4 period"
          }
        ],
        "relations": "A² = B⁴ = (AB)⁴ = 1; AB² = B²A"
      }
    },
    {
      "ordinal": 47,
      "id": "g138",
      "symbol": "c*44₂2₁ᵇ",
      "system": "Tetragonal",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*442",
        "hm": "p4m"
      },
      "kernel": {
        "orbifold": "4*2",
        "hm": "p4g"
      },
      "tos_notation": "*442/4*2",
      "book_color_signature": "*¹4²4²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *442 onto C2. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; 4-fold rotations: 0, 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *442/4*2: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 4*2. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*442/4*2 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*442/4*2"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g138",
      "image": "output/clockwork-colorings/g138.webp",
      "image_alt": "Static perfect 2-colouring for group g138: asymmetric motifs carry phase colours for Conway type *442/4*2.",
      "render": {
        "basis": [
          [
            1.000000000001,
            -0.0
          ],
          [
            0.0,
            1.000000000001
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.1089,
          0.2452
        ]
      },
      "signature_evidence": {
        "status": "type-representative",
        "label": "book-normalized representative",
        "summary": "Table 11.1 groups 2 equivalent generator signatures under the same colour type *442/4*2. The page uses the first printed short signature as a stable representative; the G/K type, not this choice of generators, is the invariant correspondence.",
        "variant_count": 2
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 16,
        "template": "cyclic_2_x_dihedral_4",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/2 a + 1/2 b",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "C",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = C² = (BC)⁴ = 1; AB = BA, AC = CA"
      }
    },
    {
      "ordinal": 48,
      "id": "g139",
      "symbol": "c4₁*2",
      "system": "Tetragonal",
      "bravais": "centred (space-time)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "4*2",
        "hm": "p4g"
      },
      "kernel": {
        "orbifold": "22×",
        "hm": "pgg"
      },
      "tos_notation": "4*2⁴/22×",
      "book_color_signature": "⁴4*²2",
      "clock_order": 4,
      "cyclic_group": "C_4",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/4",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/2",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "3/4",
          "color": "#CC79A7"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "2-fold rotations",
          "phases": [
            "0",
            "1/2"
          ]
        },
        {
          "operation": "4-fold rotations",
          "phases": [
            "1/4",
            "3/4"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/4",
            "1/2",
            "3/4"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 4*2 onto C4. The operation phases are translations: 1/2; 2-fold rotations: 0, 1/2; 4-fold rotations: 1/4, 3/4; reflections / glides: 0, 1/4, 1/2, 3/4.",
      "coloring_description": "The book-style colour type is 4*2⁴/22×: the exponent 4 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = 22×. Here [G:K] = 4 and G/K is isomorphic to C4. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/4 sends colour k to k+j modulo 4.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C4 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 4*2⁴/22×. Its prime-index layers 4*2/2*22 then 2*22/22× are tabulated; the checked phase image establishes that their extension is cyclic C4. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "4*2/2*22",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::4*2/2*22"
          },
          {
            "notation": "2*22/22×",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::2*22/22×"
          }
        ],
        "intermediate_orbifold": "2*22"
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g139",
      "image": "output/clockwork-colorings/g139.webp",
      "image_alt": "Static perfect 4-colouring for group g139: asymmetric motifs carry phase colours for Conway type 4*2⁴/22×.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.25,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.75,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.25,
              0.75
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.75,
              0.25
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.25,
              0.25
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.75,
              0.75
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.25,
              0.25
            ],
            "s": 1,
            "tau": 0.25
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.75,
              0.75
            ],
            "s": 1,
            "tau": 0.75
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.5,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.5,
              0.5
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.1327,
          0.3881
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C4 extension",
        "summary": "The book does not enumerate composite C4 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 16,
        "template": "exceptional_16",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/4-turn rotation (90°)",
            "phase": "1/4",
            "time_shift": "+1/4 period"
          }
        ],
        "relations": "A² = B⁴ = (AB)⁴ = 1; AB² = B²A"
      }
    },
    {
      "ordinal": 49,
      "id": "g224",
      "symbol": "333",
      "system": "Trigonal",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "333",
        "hm": "p3"
      },
      "kernel": {
        "orbifold": "333",
        "hm": "p3"
      },
      "tos_notation": "333",
      "book_color_signature": "333",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 333 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 333, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 333 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::333"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g224",
      "image": "output/clockwork-colorings/g224.webp",
      "image_alt": "Static perfect 1-colouring for group g224: asymmetric motifs carry phase colours for Conway type 333.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.6804,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 3,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/3-turn rotation (120°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 50,
      "id": "g225",
      "symbol": "3₂3₂3₂",
      "system": "Trigonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "333",
        "hm": "p3"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "333³/◦",
      "book_color_signature": "³3³3³3",
      "clock_order": 3,
      "cyclic_group": "C_3",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/3",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "2/3",
          "color": "#009E73"
        }
      ],
      "phase_profile": [
        {
          "operation": "3-fold rotations",
          "phases": [
            "1/3",
            "2/3"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 333 onto C3. The operation phases are 3-fold rotations: 1/3, 2/3.",
      "coloring_description": "The book colour type is 333³/◦: the exponent 3 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = ◦. Here [G:K] = 3 and G/K is isomorphic to C3. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/3 sends colour k to k+j modulo 3.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 12.1 match",
        "summary": "333³/◦ appears as a regular cyclic case in Table 12.1. Here the stabilizer H of one colour equals the all-colours kernel K.",
        "references": [
          {
            "label": "Table 12.1 · threefold color types",
            "role": "primary",
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::333³/◦"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": "g226",
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g225",
      "image": "output/clockwork-colorings/g225.webp",
      "image_alt": "Static perfect 3-colouring for group g225: asymmetric motifs carry phase colours for Conway type 333³/◦.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.6804,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "exact Table 12.1 short signature",
        "summary": "Table 12.1 prints this order-only short signature for the cited type."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 3,
        "template": "cyclic_3",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/3-turn rotation (120°)",
            "phase": "1/3",
            "time_shift": "+1/3 period"
          }
        ],
        "relations": "A³ = 1"
      }
    },
    {
      "ordinal": 51,
      "id": "g226",
      "symbol": "3₁3₁3₁",
      "system": "Trigonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "333",
        "hm": "p3"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "333³/◦",
      "book_color_signature": "³3³3³3",
      "clock_order": 3,
      "cyclic_group": "C_3",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/3",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "2/3",
          "color": "#009E73"
        }
      ],
      "phase_profile": [
        {
          "operation": "3-fold rotations",
          "phases": [
            "1/3",
            "2/3"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 333 onto C3. The operation phases are 3-fold rotations: 1/3, 2/3.",
      "coloring_description": "The book colour type is 333³/◦: the exponent 3 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = ◦. Here [G:K] = 3 and G/K is isomorphic to C3. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/3 sends colour k to k+j modulo 3.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 12.1 match",
        "summary": "333³/◦ appears as a regular cyclic case in Table 12.1. Here the stabilizer H of one colour equals the all-colours kernel K.",
        "references": [
          {
            "label": "Table 12.1 · threefold color types",
            "role": "primary",
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::333³/◦"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": "g225",
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g226",
      "image": "output/clockwork-colorings/g226.webp",
      "image_alt": "Static perfect 3-colouring for group g226: asymmetric motifs carry phase colours for Conway type 333³/◦.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.6804,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "exact Table 12.1 short signature",
        "summary": "Table 12.1 prints this order-only short signature for the cited type."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 3,
        "template": "cyclic_3",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/3-turn rotation (120°)",
            "phase": "2/3",
            "time_shift": "+2/3 period"
          }
        ],
        "relations": "A³ = 1"
      }
    },
    {
      "ordinal": 52,
      "id": "g227",
      "symbol": "r33₁3₂",
      "system": "Trigonal",
      "bravais": "R (1/3-stacked)",
      "product": false,
      "symmorphic": true,
      "parent": {
        "orbifold": "333",
        "hm": "p3"
      },
      "kernel": {
        "orbifold": "333",
        "hm": "p3"
      },
      "tos_notation": "333³/333",
      "book_color_signature": "³3³3¹3",
      "clock_order": 3,
      "cyclic_group": "C_3",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/3",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "2/3",
          "color": "#009E73"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0",
            "1/3",
            "2/3"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 333 onto C3. The operation phases are translations: 1/3, 2/3; 3-fold rotations: 0, 1/3, 2/3.",
      "coloring_description": "The book colour type is 333³/333: the exponent 3 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = 333. Here [G:K] = 3 and G/K is isomorphic to C3. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/3 sends colour k to k+j modulo 3.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 12.1 match",
        "summary": "333³/333 appears as a regular cyclic case in Table 12.1. Here the stabilizer H of one colour equals the all-colours kernel K.",
        "references": [
          {
            "label": "Table 12.1 · threefold color types",
            "role": "primary",
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::333³/333"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g227",
      "image": "output/clockwork-colorings/g227.webp",
      "image_alt": "Static perfect 3-colouring for group g227: asymmetric motifs carry phase colours for Conway type 333³/333.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          }
        ],
        "base": [
          0.4423,
          0.2214
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "exact Table 12.1 short signature",
        "summary": "Table 12.1 prints this order-only short signature for the cited type."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 9,
        "template": "elementary_3_2",
        "generators": [
          {
            "name": "A",
            "kind": "translation",
            "operation": "Translation by 1/3 a − 1/3 b",
            "phase": "2/3",
            "time_shift": "+2/3 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/3-turn rotation (120°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A³ = B³ = 1; AB = BA"
      }
    },
    {
      "ordinal": 53,
      "id": "g230",
      "symbol": "*333",
      "system": "Trigonal",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "*333",
        "hm": "p3m1"
      },
      "kernel": {
        "orbifold": "*333",
        "hm": "p3m1"
      },
      "tos_notation": "*333",
      "book_color_signature": "*333",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold *333 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group *333, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists *333 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::*333"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g230",
      "image": "output/clockwork-colorings/g230.webp",
      "image_alt": "Static perfect 1-colouring for group g230: asymmetric motifs carry phase colours for Conway type *333.",
      "render": {
        "basis": [
          [
            0.5,
            -0.866025403784
          ],
          [
            0.5,
            0.866025403784
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.6804,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 54,
      "id": "g231",
      "symbol": "*~3~3~3",
      "system": "Trigonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*333",
        "hm": "p3m1"
      },
      "kernel": {
        "orbifold": "333",
        "hm": "p3"
      },
      "tos_notation": "*333/333",
      "book_color_signature": "*²3²3²3",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *333 onto C2. The operation phases are 3-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is *333/333: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 333. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*333/333 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*333/333"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g231",
      "image": "output/clockwork-colorings/g231.webp",
      "image_alt": "Static perfect 2-colouring for group g231: asymmetric motifs carry phase colours for Conway type *333/333.",
      "render": {
        "basis": [
          [
            0.5,
            -0.866025403784
          ],
          [
            0.5,
            0.866025403784
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.6804,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "dihedral_3",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = (AB)³ = 1"
      }
    },
    {
      "ordinal": 55,
      "id": "g232",
      "symbol": "3*3",
      "system": "Trigonal",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "3*3",
        "hm": "p31m"
      },
      "kernel": {
        "orbifold": "3*3",
        "hm": "p31m"
      },
      "tos_notation": "3*3",
      "book_color_signature": "3*3",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 3*3 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 3*3, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 3*3 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::3*3"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g232",
      "image": "output/clockwork-colorings/g232.webp",
      "image_alt": "Static perfect 1-colouring for group g232: asymmetric motifs carry phase colours for Conway type 3*3.",
      "render": {
        "basis": [
          [
            0.866025403784,
            0.499999999999
          ],
          [
            -0.866025403784,
            0.5
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.5851,
          0.7928
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 56,
      "id": "g233",
      "symbol": "3*~3",
      "system": "Trigonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "3*3",
        "hm": "p31m"
      },
      "kernel": {
        "orbifold": "333",
        "hm": "p3"
      },
      "tos_notation": "3*3/333",
      "book_color_signature": "¹3*²3",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 3*3 onto C2. The operation phases are 3-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is 3*3/333: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 333. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "3*3/333 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::3*3/333"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g233",
      "image": "output/clockwork-colorings/g233.webp",
      "image_alt": "Static perfect 2-colouring for group g233: asymmetric motifs carry phase colours for Conway type 3*3/333.",
      "render": {
        "basis": [
          [
            0.866025403784,
            0.499999999999
          ],
          [
            -0.866025403784,
            0.5
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.5851,
          0.7928
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "dihedral_3",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = (AB)³ = 1"
      }
    },
    {
      "ordinal": 57,
      "id": "g234",
      "symbol": "r3₂*3",
      "system": "Trigonal",
      "bravais": "R (1/3-stacked)",
      "product": false,
      "symmorphic": true,
      "parent": {
        "orbifold": "3*3",
        "hm": "p31m"
      },
      "kernel": {
        "orbifold": "*333",
        "hm": "p3m1"
      },
      "tos_notation": "3*3³/*333",
      "book_color_signature": "³3*¹3",
      "clock_order": 3,
      "cyclic_group": "C_3",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/3",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "2/3",
          "color": "#009E73"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0",
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/3",
            "2/3"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 3*3 onto C3. The operation phases are translations: 1/3, 2/3; 3-fold rotations: 0, 1/3, 2/3; reflections / glides: 0, 1/3, 2/3.",
      "coloring_description": "The book colour type is 3*3³/*333: the exponent 3 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = *333. Here [G:K] = 3 and G/K is isomorphic to C3. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/3 sends colour k to k+j modulo 3.",
      "book_audit": {
        "status": "internal-discrepancy",
        "status_label": "book-internal discrepancy",
        "summary": "No one book row contains both the displayed short generator signature and the computed kernel. Table 12.1 and the derivation on p. 158 pair the short form ³3*¹3 with type 3*3³/◦ (the exponent 3 is understood there), while Table 13.1 prints the computed type 3*3³/*333 beside a differently positioned full cycle label. The page keeps the kernel computed from the clock operations and links Frank Farris's independent p31m/3p3m1 construction as a check on that parent/kernel type.",
        "references": [
          {
            "label": "Table 13.1 · later primefold summary",
            "role": "primary",
            "printed_page": 164,
            "pdf_page": 183,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA164",
            "excerpt_key": "p164::g234-single-slash-table"
          },
          {
            "label": "Table 12.1 · same short form, conflicting kernel",
            "role": "conflict",
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::3*3³/◦-conflict"
          },
          {
            "label": "Threefold derivation · conflicting prose",
            "role": "conflict",
            "printed_page": 158,
            "pdf_page": 177,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA158",
            "excerpt_key": "p158::g234-prose-conflict"
          }
        ],
        "independent_reference": {
          "label": "Farris, Natural Color Symmetry, p. 136",
          "url": "https://archive.bridgesmathart.org/2017/bridges2017-131.pdf#page=6"
        },
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g234",
      "image": "output/clockwork-colorings/g234.webp",
      "image_alt": "Static perfect 3-colouring for group g234: asymmetric motifs carry phase colours for Conway type 3*3³/*333.",
      "render": {
        "basis": [
          [
            0.5,
            -0.866025403784
          ],
          [
            0.5,
            0.866025403784
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          }
        ],
        "base": [
          0.7994,
          0.7928
        ]
      },
      "signature_evidence": {
        "status": "book-internal-discrepancy",
        "label": "signature and kernel split across book rows",
        "summary": "No single book row contains both the displayed order-only signature and the computed kernel *333. Page 158 supports the short signature but gives kernel ◦; Table 13.1 supports the 3*3³/*333 type but prints a differently positioned full cycle label. The page therefore states the synthesis explicitly instead of calling it a direct book signature."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 18,
        "template": "exceptional_18",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/3-turn rotation (120°)",
            "phase": "2/3",
            "time_shift": "+2/3 period"
          }
        ],
        "relations": "A² = B³ = 1; B(ABA) = (ABA)B"
      }
    },
    {
      "ordinal": 58,
      "id": "g235",
      "symbol": "r3₂*~3",
      "system": "Trigonal",
      "bravais": "R (1/3-stacked)",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "3*3",
        "hm": "p31m"
      },
      "kernel": {
        "orbifold": "333",
        "hm": "p3"
      },
      "tos_notation": "3*3⁶/333",
      "book_color_signature": "³3*²3",
      "clock_order": 6,
      "cyclic_group": "C_6",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/6",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/3",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "1/2",
          "color": "#CC79A7"
        },
        {
          "index": 4,
          "phase": "2/3",
          "color": "#D55E00"
        },
        {
          "index": 5,
          "phase": "5/6",
          "color": "#56B4E9"
        }
      ],
      "phase_profile": [
        {
          "operation": "translations",
          "phases": [
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0",
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/6",
            "1/2",
            "5/6"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 3*3 onto C6. The operation phases are translations: 1/3, 2/3; 3-fold rotations: 0, 1/3, 2/3; reflections / glides: 1/6, 1/2, 5/6.",
      "coloring_description": "The book-style colour type is 3*3⁶/333: the exponent 6 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = 333. Here [G:K] = 6 and G/K is isomorphic to C6. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/6 sends colour k to k+j modulo 6.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C6 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 3*3⁶/333. Its prime-index layers 3*3/333 then 333³/333 are tabulated; the checked phase image establishes that their extension is cyclic C6. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "3*3/333",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::3*3/333"
          },
          {
            "notation": "333³/333",
            "index": 3,
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::333³/333"
          }
        ],
        "intermediate_orbifold": "333"
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g235",
      "image": "output/clockwork-colorings/g235.webp",
      "image_alt": "Static perfect 6-colouring for group g235: asymmetric motifs carry phase colours for Conway type 3*3⁶/333.",
      "render": {
        "basis": [
          [
            0.5,
            -0.866025403784
          ],
          [
            0.5,
            0.866025403784
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.16666666666666666
          },
          {
            "M": [
              [
                -1,
                1
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              [
                0,
                1
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            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.8333333333333334
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.16666666666666666
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.8333333333333334
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.3333333333333333,
              0.6666666666666666
            ],
            "s": 1,
            "tau": 0.16666666666666666
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.6666666666666666,
              0.3333333333333333
            ],
            "s": 1,
            "tau": 0.8333333333333334
          }
        ],
        "base": [
          0.7994,
          0.7928
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C6 extension",
        "summary": "The book does not enumerate composite C6 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 18,
        "template": "exceptional_18",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "rotation",
            "operation": "1/3-turn rotation (120°)",
            "phase": "2/3",
            "time_shift": "+2/3 period"
          }
        ],
        "relations": "A² = B³ = 1; B(ABA) = (ABA)B"
      }
    },
    {
      "ordinal": 59,
      "id": "g243",
      "symbol": "632",
      "system": "Hexagonal",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "632",
        "hm": "p6"
      },
      "kernel": {
        "orbifold": "632",
        "hm": "p6"
      },
      "tos_notation": "632",
      "book_color_signature": "632",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold 632 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group 632, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists 632 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::632"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g243",
      "image": "output/clockwork-colorings/g243.webp",
      "image_alt": "Static perfect 1-colouring for group g243: asymmetric motifs carry phase colours for Conway type 632.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.728,
          0.15
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/6-turn rotation (60°)",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 60,
      "id": "g244",
      "symbol": "6₂3₂2",
      "system": "Hexagonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "632",
        "hm": "p6"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "632³/2222",
      "book_color_signature": "³6³3¹2",
      "clock_order": 3,
      "cyclic_group": "C_3",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/3",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "2/3",
          "color": "#009E73"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "1/3",
            "2/3"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 632 onto C3. The operation phases are 2-fold rotations: 0; 3-fold rotations: 1/3, 2/3; 6-fold rotations: 1/3, 2/3.",
      "coloring_description": "The book colour type is 632³/2222: the exponent 3 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 3 and G/K is isomorphic to C3. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/3 sends colour k to k+j modulo 3.",
      "book_audit": {
        "status": "internal-discrepancy",
        "status_label": "book typo resolved by pp. 157 and 164",
        "summary": "The page's ³6³3¹2 is derived in the prose on p. 157 and Table 13.1 assigns the three generators a 3-cycle, its inverse, and the identity for type 632³/2222. Table 12.1 on p. 156 instead prints ³6²3²2 beside 632/2222. Those raised 2s denote transpositions and cannot describe a regular C3 action. We therefore treat p. 156 as a book error; it is not listed in the authors' online errata.",
        "references": [
          {
            "label": "Table 13.1 · exact later signature and type",
            "role": "primary",
            "printed_page": 164,
            "pdf_page": 183,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA164",
            "excerpt_key": "p164::632³/2222-exact"
          },
          {
            "label": "Threefold derivation · correct short signature",
            "role": "supporting",
            "printed_page": 157,
            "pdf_page": 176,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA157",
            "excerpt_key": "p157::632-regular-derivation"
          },
          {
            "label": "Table 12.1 · conflicting earlier short signature",
            "role": "conflict",
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::632³/2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": "g245",
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g244",
      "image": "output/clockwork-colorings/g244.webp",
      "image_alt": "Static perfect 3-colouring for group g244: asymmetric motifs carry phase colours for Conway type 632³/2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.728,
          0.15
        ]
      },
      "signature_evidence": {
        "status": "book-internal-discrepancy",
        "label": "p. 156 typo corrected by pp. 157 and 164",
        "summary": "The displayed ³6³3¹2 is derived on p. 157 and printed with full permutations in Table 13.1. Table 12.1 instead has the inconsistent ³6²3²2; the official errata does not list that typo."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "cyclic_6",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/6-turn rotation (60°)",
            "phase": "2/3",
            "time_shift": "+2/3 period"
          }
        ],
        "relations": "A⁶ = 1"
      }
    },
    {
      "ordinal": 61,
      "id": "g245",
      "symbol": "6₄3₁2",
      "system": "Hexagonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "632",
        "hm": "p6"
      },
      "kernel": {
        "orbifold": "2222",
        "hm": "p2"
      },
      "tos_notation": "632³/2222",
      "book_color_signature": "³6³3¹2",
      "clock_order": 3,
      "cyclic_group": "C_3",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/3",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "2/3",
          "color": "#009E73"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "1/3",
            "2/3"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 632 onto C3. The operation phases are 2-fold rotations: 0; 3-fold rotations: 1/3, 2/3; 6-fold rotations: 1/3, 2/3.",
      "coloring_description": "The book colour type is 632³/2222: the exponent 3 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = 2222. Here [G:K] = 3 and G/K is isomorphic to C3. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/3 sends colour k to k+j modulo 3.",
      "book_audit": {
        "status": "internal-discrepancy",
        "status_label": "book typo resolved by pp. 157 and 164",
        "summary": "The page's ³6³3¹2 is derived in the prose on p. 157 and Table 13.1 assigns the three generators a 3-cycle, its inverse, and the identity for type 632³/2222. Table 12.1 on p. 156 instead prints ³6²3²2 beside 632/2222. Those raised 2s denote transpositions and cannot describe a regular C3 action. We therefore treat p. 156 as a book error; it is not listed in the authors' online errata.",
        "references": [
          {
            "label": "Table 13.1 · exact later signature and type",
            "role": "primary",
            "printed_page": 164,
            "pdf_page": 183,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA164",
            "excerpt_key": "p164::632³/2222-exact"
          },
          {
            "label": "Threefold derivation · correct short signature",
            "role": "supporting",
            "printed_page": 157,
            "pdf_page": 176,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA157",
            "excerpt_key": "p157::632-regular-derivation"
          },
          {
            "label": "Table 12.1 · conflicting earlier short signature",
            "role": "conflict",
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::632³/2222"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": "g244",
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g245",
      "image": "output/clockwork-colorings/g245.webp",
      "image_alt": "Static perfect 3-colouring for group g245: asymmetric motifs carry phase colours for Conway type 632³/2222.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.728,
          0.15
        ]
      },
      "signature_evidence": {
        "status": "book-internal-discrepancy",
        "label": "p. 156 typo corrected by pp. 157 and 164",
        "summary": "The displayed ³6³3¹2 is derived on p. 157 and printed with full permutations in Table 13.1. Table 12.1 instead has the inconsistent ³6²3²2; the official errata does not list that typo."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "cyclic_6",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/6-turn rotation (60°)",
            "phase": "1/3",
            "time_shift": "+1/3 period"
          }
        ],
        "relations": "A⁶ = 1"
      }
    },
    {
      "ordinal": 62,
      "id": "g246",
      "symbol": "6₃32₁",
      "system": "Hexagonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "632",
        "hm": "p6"
      },
      "kernel": {
        "orbifold": "333",
        "hm": "p3"
      },
      "tos_notation": "632/333",
      "book_color_signature": "²6¹3²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 632 onto C2. The operation phases are 2-fold rotations: 1/2; 3-fold rotations: 0; 6-fold rotations: 1/2.",
      "coloring_description": "The book colour type is 632/333: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 333. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "632/333 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::632/333"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g246",
      "image": "output/clockwork-colorings/g246.webp",
      "image_alt": "Static perfect 2-colouring for group g246: asymmetric motifs carry phase colours for Conway type 632/333.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.728,
          0.15
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "cyclic_6",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/6-turn rotation (60°)",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A⁶ = 1"
      }
    },
    {
      "ordinal": 63,
      "id": "g247",
      "symbol": "6₅3₂2₁",
      "system": "Hexagonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "632",
        "hm": "p6"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "632⁶/◦",
      "book_color_signature": "⁶6³3²2",
      "clock_order": 6,
      "cyclic_group": "C_6",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/6",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/3",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "1/2",
          "color": "#CC79A7"
        },
        {
          "index": 4,
          "phase": "2/3",
          "color": "#D55E00"
        },
        {
          "index": 5,
          "phase": "5/6",
          "color": "#56B4E9"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "1/6",
            "5/6"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 632 onto C6. The operation phases are 2-fold rotations: 1/2; 3-fold rotations: 1/3, 2/3; 6-fold rotations: 1/6, 5/6.",
      "coloring_description": "The book-style colour type is 632⁶/◦: the exponent 6 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = ◦. Here [G:K] = 6 and G/K is isomorphic to C6. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/6 sends colour k to k+j modulo 6.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C6 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 632⁶/◦. Its prime-index layers 632/333 then 333³/◦ are tabulated; the checked phase image establishes that their extension is cyclic C6. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "632/333",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::632/333"
          },
          {
            "notation": "333³/◦",
            "index": 3,
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::333³/◦"
          }
        ],
        "intermediate_orbifold": "333"
      },
      "inverse_clock_mate": "g248",
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g247",
      "image": "output/clockwork-colorings/g247.webp",
      "image_alt": "Static perfect 6-colouring for group g247: asymmetric motifs carry phase colours for Conway type 632⁶/◦.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.16666666666666666
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.8333333333333334
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.728,
          0.15
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C6 extension",
        "summary": "The book does not enumerate composite C6 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "cyclic_6",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/6-turn rotation (60°)",
            "phase": "1/6",
            "time_shift": "+1/6 period"
          }
        ],
        "relations": "A⁶ = 1"
      }
    },
    {
      "ordinal": 64,
      "id": "g248",
      "symbol": "6₁3₁2₁",
      "system": "Hexagonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "632",
        "hm": "p6"
      },
      "kernel": {
        "orbifold": "◦",
        "hm": "p1"
      },
      "tos_notation": "632⁶/◦",
      "book_color_signature": "⁶6³3²2",
      "clock_order": 6,
      "cyclic_group": "C_6",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/6",
          "color": "#E69F00"
        },
        {
          "index": 2,
          "phase": "1/3",
          "color": "#009E73"
        },
        {
          "index": 3,
          "phase": "1/2",
          "color": "#CC79A7"
        },
        {
          "index": 4,
          "phase": "2/3",
          "color": "#D55E00"
        },
        {
          "index": 5,
          "phase": "5/6",
          "color": "#56B4E9"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "1/3",
            "2/3"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "1/6",
            "5/6"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold 632 onto C6. The operation phases are 2-fold rotations: 1/2; 3-fold rotations: 1/3, 2/3; 6-fold rotations: 1/6, 5/6.",
      "coloring_description": "The book-style colour type is 632⁶/◦: the exponent 6 on G counts the colours, and the orbifold signature after the slash is the colour-fixing kernel K = ◦. Here [G:K] = 6 and G/K is isomorphic to C6. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/6 sends colour k to k+j modulo 6.",
      "book_audit": {
        "status": "composite-extension",
        "status_label": "regular cyclic C6 extension",
        "summary": "The book stops after primefold enumeration, so it does not list 632⁶/◦. Its prime-index layers 632/333 then 333³/◦ are tabulated; the checked phase image establishes that their extension is cyclic C6. The superscripted notation follows the rule on p. 155.",
        "references": [
          {
            "label": "Gⁿ/H/K notation and slash rule",
            "role": "primary",
            "printed_page": 155,
            "pdf_page": 174,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA155",
            "excerpt_key": "p155::slash-rule"
          },
          {
            "label": "End of the book's primefold enumeration",
            "role": "scope",
            "printed_page": 169,
            "pdf_page": 188,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA169",
            "excerpt_key": "p169::primefold-scope"
          }
        ],
        "prime_chain": [
          {
            "notation": "632/333",
            "index": 2,
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::632/333"
          },
          {
            "notation": "333³/◦",
            "index": 3,
            "printed_page": 156,
            "pdf_page": 175,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA156",
            "excerpt_key": "p156::333³/◦"
          }
        ],
        "intermediate_orbifold": "333"
      },
      "inverse_clock_mate": "g247",
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g248",
      "image": "output/clockwork-colorings/g248.webp",
      "image_alt": "Static perfect 6-colouring for group g248: asymmetric motifs carry phase colours for Conway type 632⁶/◦.",
      "render": {
        "basis": [
          [
            1,
            0
          ],
          [
            -0.5,
            0.8660254037844386
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.6666666666666666
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.3333333333333333
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.8333333333333334
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.16666666666666666
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.728,
          0.15
        ]
      },
      "signature_evidence": {
        "status": "rule-extension",
        "label": "Goodman–Strauss-style C6 extension",
        "summary": "The book does not enumerate composite C6 colourings. This short signature is derived from its rule: replace each generator permutation by its order."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 6,
        "template": "cyclic_6",
        "generators": [
          {
            "name": "A",
            "kind": "rotation",
            "operation": "1/6-turn rotation (60°)",
            "phase": "5/6",
            "time_shift": "+5/6 period"
          }
        ],
        "relations": "A⁶ = 1"
      }
    },
    {
      "ordinal": 65,
      "id": "g268",
      "symbol": "*632",
      "system": "Hexagonal",
      "bravais": "P",
      "product": true,
      "symmorphic": true,
      "parent": {
        "orbifold": "*632",
        "hm": "p6m"
      },
      "kernel": {
        "orbifold": "*632",
        "hm": "p6m"
      },
      "tos_notation": "*632",
      "book_color_signature": "*632",
      "clock_order": 1,
      "cyclic_group": "C_1",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0"
          ]
        }
      ],
      "clockwork_description": "The direct-product lift over plane orbifold *632 has trivial phase character: every spatial operation maps to phase 0.",
      "coloring_description": "The phase character is trivial, so K = G. In the book's terminology this is the onefold plane group *632, not a G¹/G colour-type label. The static plate is monochrome.",
      "book_audit": {
        "status": "plane-group",
        "status_label": "ordinary plane-group table",
        "summary": "Table 3.2 lists *632 among the 17 plane groups. The book calls this a onefold coloring; it does not write a G¹/G color type.",
        "references": [
          {
            "label": "Table 3.2 · the 17 plane groups",
            "role": "primary",
            "printed_page": 40,
            "pdf_page": 59,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA40",
            "excerpt_key": "p40::*632"
          },
          {
            "label": "Onefold and n-fold colorings",
            "role": "supporting",
            "printed_page": 153,
            "pdf_page": 172,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA153",
            "excerpt_key": "p153::onefold-nfold-definition"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g268",
      "image": "output/clockwork-colorings/g268.webp",
      "image_alt": "Static perfect 1-colouring for group g268: asymmetric motifs carry phase colours for Conway type *632.",
      "render": {
        "basis": [
          [
            0.5,
            -0.866025403784
          ],
          [
            0.5,
            0.866025403785
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.5613,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "onefold",
        "label": "ordinary onefold plane group",
        "summary": "No nontrivial short colour signature is needed."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 12,
        "template": "omitted_trivial_time_action",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "omitted"
      }
    },
    {
      "ordinal": 66,
      "id": "g269",
      "symbol": "*6₃~3~2₁",
      "system": "Hexagonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*632",
        "hm": "p6m"
      },
      "kernel": {
        "orbifold": "3*3",
        "hm": "p31m"
      },
      "tos_notation": "*632/3*3",
      "book_color_signature": "*¹6²3²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *632 onto C2. The operation phases are 2-fold rotations: 1/2; 3-fold rotations: 0; 6-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *632/3*3: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 3*3. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*632/3*3 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*632/3*3"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g269",
      "image": "output/clockwork-colorings/g269.webp",
      "image_alt": "Static perfect 2-colouring for group g269: asymmetric motifs carry phase colours for Conway type *632/3*3.",
      "render": {
        "basis": [
          [
            0.866025403784,
            0.499999999999
          ],
          [
            -0.866025403784,
            0.5
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.5613,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 12,
        "template": "dihedral_6",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "0",
            "time_shift": "none"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = (AB)⁶ = 1"
      }
    },
    {
      "ordinal": 67,
      "id": "g270",
      "symbol": "*~6~3~2",
      "system": "Hexagonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*632",
        "hm": "p6m"
      },
      "kernel": {
        "orbifold": "632",
        "hm": "p6"
      },
      "tos_notation": "*632/632",
      "book_color_signature": "*²6²3²2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *632 onto C2. The operation phases are 2-fold rotations: 0; 3-fold rotations: 0; 6-fold rotations: 0; reflections / glides: 1/2.",
      "coloring_description": "The book colour type is *632/632: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = 632. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*632/632 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*632/632"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g270",
      "image": "output/clockwork-colorings/g270.webp",
      "image_alt": "Static perfect 2-colouring for group g270: asymmetric motifs carry phase colours for Conway type *632/632.",
      "render": {
        "basis": [
          [
            0.5,
            -0.866025403784
          ],
          [
            0.5,
            0.866025403785
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          }
        ],
        "base": [
          0.5613,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 12,
        "template": "dihedral_6",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          }
        ],
        "relations": "A² = B² = (AB)⁶ = 1"
      }
    },
    {
      "ordinal": 68,
      "id": "g271",
      "symbol": "*~6₃32₁",
      "system": "Hexagonal",
      "bravais": "P",
      "product": false,
      "symmorphic": false,
      "parent": {
        "orbifold": "*632",
        "hm": "p6m"
      },
      "kernel": {
        "orbifold": "*333",
        "hm": "p3m1"
      },
      "tos_notation": "*632/*333",
      "book_color_signature": "*²6¹3¹2",
      "clock_order": 2,
      "cyclic_group": "C_2",
      "phase_residues": [
        {
          "index": 0,
          "phase": "0",
          "color": "#0072B2"
        },
        {
          "index": 1,
          "phase": "1/2",
          "color": "#E69F00"
        }
      ],
      "phase_profile": [
        {
          "operation": "2-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "3-fold rotations",
          "phases": [
            "0"
          ]
        },
        {
          "operation": "6-fold rotations",
          "phases": [
            "1/2"
          ]
        },
        {
          "operation": "reflections / glides",
          "phases": [
            "0",
            "1/2"
          ]
        }
      ],
      "clockwork_description": "The phase character maps the plane orbifold *632 onto C2. The operation phases are 2-fold rotations: 1/2; 3-fold rotations: 0; 6-fold rotations: 1/2; reflections / glides: 0, 1/2.",
      "coloring_description": "The book colour type is *632/*333: twofold is understood, so no exponent 2 is printed, and the orbifold signature after the slash is the colour-fixing kernel K = *333. Here [G:K] = 2 and G/K is isomorphic to C2. The one-slash form is valid because this cyclic colour action is regular, so the stabilizer H of one chosen colour equals K. An operation at phase j/2 sends colour k to k+j modulo 2.",
      "book_audit": {
        "status": "direct-table",
        "status_label": "direct Table 11.1 match",
        "summary": "*632/*333 appears directly in Table 11.1. Its single slash is the book's G/K notation for a regular two-colour action.",
        "references": [
          {
            "label": "Table 11.1 · twofold color types",
            "role": "primary",
            "printed_page": 140,
            "pdf_page": 159,
            "url": "https://books.google.com/books?id=EtQCk0TNafsC&pg=PA140",
            "excerpt_key": "p140::*632/*333"
          }
        ],
        "prime_chain": []
      },
      "inverse_clock_mate": null,
      "catalog_url": "https://yaroslavvb.github.io/animated-groups-fable/catalog.html?time=forward#g271",
      "image": "output/clockwork-colorings/g271.webp",
      "image_alt": "Static perfect 2-colouring for group g271: asymmetric motifs carry phase colours for Conway type *632/*333.",
      "render": {
        "basis": [
          [
            0.5,
            -0.866025403784
          ],
          [
            0.5,
            0.866025403784
          ]
        ],
        "ops": [
          {
            "M": [
              [
                -1,
                0
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                0
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                -1,
                1
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                -1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                -1
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                -1,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                0,
                1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                0,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                -1
              ],
              [
                1,
                0
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.5
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                0,
                1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          },
          {
            "M": [
              [
                1,
                0
              ],
              [
                1,
                -1
              ]
            ],
            "v": [
              0.0,
              0.0
            ],
            "s": 1,
            "tau": 0.0
          }
        ],
        "base": [
          0.5613,
          0.6738
        ]
      },
      "signature_evidence": {
        "status": "exact-printed",
        "label": "unique Table 11.1 short signature",
        "summary": "Table 11.1 prints one short generator signature for this colour type, and the page reproduces it."
      },
      "cell_action_presentation": {
        "quotient": "G/Λ",
        "quotient_order": 12,
        "template": "dihedral_6",
        "generators": [
          {
            "name": "A",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 1",
            "phase": "1/2",
            "time_shift": "+1/2 period"
          },
          {
            "name": "B",
            "kind": "mirror",
            "operation": "Mirror reflection in axis direction 2",
            "phase": "0",
            "time_shift": "none"
          }
        ],
        "relations": "A² = B² = (AB)⁶ = 1"
      }
    }
  ]
}