{ "meta": { "schema_version": 3, "title": "Cyclic colourings and polar space groups", "source": "data/clockwork-coloring-correspondence.json", "source_sha256": "242a467001ac496aaf048ad2467886f79e5e6139630789ead537ef76bcae1330", "source_groups": 68, "wallpaper_families": 17, "space_group_types": 68, "displayed_nontrivial_groups": 51, "displayed_wallpaper_families": 14, "omitted_trivial_products": 17, "space_group_numbering": "International Tables for Crystallography", "mapping_database": "Pinned spglib 2.6.0 space-group type results", "construction": "Embed (M, v, τ) as (diag(M, 1), (v, τ)); cyclic phase becomes fractional height along the new z coordinate.", "scope_caveat": "This 68-record forward regular-cyclic subset corresponds exactly to the 68 polar three-dimensional space-group types. General coloured wallpaper groups are broader, and there is no one-to-one correspondence between all coloured wallpaper groups and all 230 space groups.", "image_size": [ 720, 480 ], "image_note": "Space-group plate hues repeat the cyclic phase palette only to trace the height lift; colour is not additional crystallographic structure in the 3D group.", "external_reference": "Jeremy K. Cockcroft, A Hypertext Book of Crystallographic Space Group Diagrams and Tables, UCL/Birkbeck College, 1997-1999.", "external_reference_index": "http://img.chem.ucl.ac.uk/sgp/large/sgp.htm", "external_reference_cache_policy": "The project does not redistribute UCL pages or images because their published end-user licence prohibits Internet distribution; hover cards reuse locally generated project plates.", "polar_it_numbers": [ 1, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 75, 76, 77, 78, 79, 80, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 143, 144, 145, 146, 156, 157, 158, 159, 160, 161, 168, 169, 170, 171, 172, 173, 183, 184, 185, 186 ], "family_counts": { "p1": 1, "p2": 3, "pm": 1, "pg": 1, "cm": 2, "pmm": 6, "pmg": 6, "pgg": 4, "cmm": 6, "p4": 6, "p4m": 6, "p4g": 6, "p3": 4, "p3m1": 2, "p31m": 4, "p6": 6, "p6m": 4 } }, "groups": [ { "ordinal": 1, "id": "g1", "symbol": "o", "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": [], "cell_action_presentation": { "quotient": "G/Λ", "quotient_order": 1, "template": "omitted_trivial_time_action", "generators": [], "relations": "omitted" }, "space_group_presentation": 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 ] }, "lift_formula": "(x, y, z) ↦ (M(x, y) + v, z + τ)", "lift_operations": [ { "R": [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], "t": [ 0.0, 0.0, 0.0 ] } ], "space_group": { "it_number": 1, "hm_short": "P1", "hm_full": "P 1", "hall": "P 1", "choice": "", "crystal_system": "triclinic", "point_group": "1", "polar_axis": "z (constructed lift coordinate; before conventional-setting normalization)", "image": "output/space-groups/g1.webp", "image_alt": "Axonometric unit-cell view of polar space group No. 1 P1, lifted from cyclic colouring g1.", "ucl_reference_url": "http://img.chem.ucl.ac.uk/sgp/large/001az1.htm", "reference_preview_image": "output/space-groups/g1.webp", "reference_preview_note": "Locally cached project rendering; the linked UCL diagram is not redistributed because its published licence prohibits Internet distribution of copies." } }, { "ordinal": 2, "id": "g5", "symbol": "2222", "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" ] } ], "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" }, "space_group_presentation": 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 ] }, "lift_formula": "(x, y, z) ↦ (M(x, y) + v, z + τ)", "lift_operations": [ { "R": [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, 1 ] ], "t": [ 0.0, 0.0, 0.0 ] }, { "R": [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], "t": [ 0.0, 0.0, 0.0 ] } ], "space_group": { "it_number": 3, "hm_short": "P2", "hm_full": "P 1 2 1", "hall": "P 2y", "choice": "b", "crystal_system": "monoclinic", "point_group": "2", "polar_axis": "z (constructed lift coordinate; before conventional-setting normalization)", "image": "output/space-groups/g5.webp", "image_alt": "Axonometric unit-cell view of polar space group No. 3 P2, lifted from cyclic colouring g5.", "ucl_reference_url": "http://img.chem.ucl.ac.uk/sgp/large/003ay1.htm", "reference_preview_image": "output/space-groups/g5.webp", "reference_preview_note": "Locally cached project rendering; the linked UCL diagram is not redistributed because its published licence prohibits Internet distribution of copies." } }, { "ordinal": 3, "id": "g6", "symbol": "2₁2₁2₁2₁", "parent": { "orbifold": "2222", "hm": "p2" }, "kernel": { 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"P2_1", "hm_full": "P 1 2_1 1", "hall": "P 2yb", "choice": "b", "crystal_system": "monoclinic", "point_group": "2", "polar_axis": "z (constructed lift coordinate; before conventional-setting normalization)", "image": "output/space-groups/g6.webp", "image_alt": "Axonometric unit-cell view of polar space group No. 4 P2_1, lifted from cyclic colouring g6.", "ucl_reference_url": "http://img.chem.ucl.ac.uk/sgp/large/004ay1.htm", "reference_preview_image": "output/space-groups/g6.webp", "reference_preview_note": "Locally cached project rendering; the linked UCL diagram is not redistributed because its published licence prohibits Internet distribution of copies." } }, { "ordinal": 4, "id": "g7", "symbol": "c222₁2₁", "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, 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copies." } }, { "ordinal": 5, "id": "g8", "symbol": "*×", "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" ] } ], "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" }, "space_group_presentation": 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, 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