Spacetime Groups

Clockwork/colouring correspondence wallpaper group 16 of 17

632 p6 6 forward groups

A sphere with cone points of orders 6, 3, and 2.

  1. 363312

    p. 156 typo corrected by pp. 157 and 164

    62322
    Static perfect 3-colouring for group g244: asymmetric motifs carry phase colours for Conway type 632³/2222.
    Space–time lift diagram for g244 and polar space group No. 171 P6_2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Sr(S2O6)(H2O)4.
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    3D crystal · Sr(S2O6)(H2O)4No. 171 P62 CrystalSymmetry exampleOpen model on Sketchfab
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    GeneratorColorTime
    αone-sixth turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C3+1/3 period
    βone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C32+2/3 period
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone

    Relations Γ = ⟨α, β, γ | α⁶ = β³ = γ² = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 164Short-form evidenceconflicting row 362322 · p. 156
    • Other names(623220)P62No. 171C64P 62g245g244
    • Vladimir catalog632/2222[3]
    Extra links

  2. 363312

    p. 156 typo corrected by pp. 157 and 164

    64312
    Static perfect 3-colouring for group g245: asymmetric motifs carry phase colours for Conway type 632³/2222.
    Space–time lift diagram for g245 and polar space group No. 172 P6_4; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Ca(S2O6)(H2O)4.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Ca(S2O6)(H2O)4No. 172 P64 CrystalSymmetry exampleOpen model on Sketchfab
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    Presentation

    GeneratorColorTime
    αone-sixth turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C32+2/3 period
    βone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C3+1/3 period
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone

    Relations Γ = ⟨α, β, γ | α⁶ = β³ = γ² = αβγ = 1⟩

    Clock normalization: A, B, … stay in forward phase order. The book’s independently lettered cycles are available in the source audit; the short-form orders are unchanged.

    Book cycle labels
    • α (ABC)
    • β (ACB)

    The source letters are an independent simultaneous relabelling; they do not change the fixed phase colors above.

    Identifications

    • The Symmetries of Things · p. 164Short-form evidenceconflicting row 362322 · p. 156
    • Other names(623220)P64No. 172C65P 64g244g245
    • Vladimir catalog632/2222[3]
    Extra links

  3. 261322

    63321
    Static perfect 2-colouring for group g246: asymmetric motifs carry phase colours for Conway type 632/333.
    Space–time lift diagram for g246 and polar space group No. 173 P6_3; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Nepheline.
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    3D crystal · NephelineNo. 173 P63 CrystalSymmetry exampleOpen model on Sketchfab
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    Presentation

    GeneratorColorTime
    αone-sixth turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    βone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period

    Relations Γ = ⟨α, β, γ | α⁶ = β³ = γ² = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(633021)P63No. 173C66P 6cg246
    Extra links

  4. 663322

    Derived C6 short form — no literal book row

    653221
    Static perfect 6-colouring for group g247: asymmetric motifs carry phase colours for Conway type 632⁶/◦.
    Space–time lift diagram for g247 and polar space group No. 170 P6_5; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine NaCoPO4.
    Opens 3Dmol.js with the open COD structure externally. Only one crystal viewer stays live at a time.
    3D crystal · NaCoPO4No. 170 P65 CrystalSymmetry exampleCOD 2101517 structure data
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    GeneratorColorTime
    αone-sixth turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C65+5/6 period
    βone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C64+2/3 period
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C63+1/2 period

    Relations Γ = ⟨α, β, γ | α⁶ = β³ = γ² = αβγ = 1⟩

    Cyclic extension: these clock powers come from the audited time shifts; the book does not print a composite-colour row.

    Derived source cycle labels
    • α (ABDFEC)
    • β (ADE)(BFC)
    • γ (AF)(BE)(CD)

    The source letters are an independent simultaneous relabelling; they do not change the fixed phase colors above.

    Identifications

    • The Symmetries of Things · p. 155no literal row; short form derivedShort-form evidenceC2 layer 261322 · p. 140C3 layer 333333 · p. 156
    • Other names(613121)P65No. 170C63P 65g248g247
    Extra links

  5. 663322

    Derived C6 short form — no literal book row

    613121
    Static perfect 6-colouring for group g248: asymmetric motifs carry phase colours for Conway type 632⁶/◦.
    Space–time lift diagram for g248 and polar space group No. 169 P6_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Al2S3.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Al2S3No. 169 P61 CrystalSymmetry exampleOpen model on Sketchfab
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    Presentation

    GeneratorColorTime
    αone-sixth turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C6+1/6 period
    βone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C62+1/3 period
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C63+1/2 period

    Relations Γ = ⟨α, β, γ | α⁶ = β³ = γ² = αβγ = 1⟩

    Cyclic extension: these clock powers come from the audited time shifts; the book does not print a composite-colour row.

    Derived source cycle labels
    • α (ABDFEC)
    • β (ADE)(BFC)
    • γ (AF)(BE)(CD)

    The source letters are an independent simultaneous relabelling; they do not change the fixed phase colors above.

    Identifications

    • The Symmetries of Things · p. 155no literal row; short form derivedShort-form evidenceC2 layer 261322 · p. 140C3 layer 333333 · p. 156
    • Other names(613121)P61No. 169C62P 61g247g248
    Extra links

  6. 632

    632
    Static perfect 1-colouring for group g243: asymmetric motifs carry phase colours for Conway type 632.
    Space–time lift diagram for g243 and polar space group No. 168 P6; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine K2Ta4F4O9.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · K2Ta4F4O9No. 168 P6 CrystalSymmetry exampleOpen model on Sketchfab
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    Presentation

    GeneratorColorTime
    αone-sixth turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone

    Relations Γ = ⟨α, β, γ | α⁶ = β³ = γ² = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(603020)P6No. 168C61P 6g243
    Extra links

Data

68-record correspondence JSON · 68-record crystal-example map · Vladimir Bulatov’s catalog of colour groups · 68-record catalog link map · generator · book-excerpt map · errata · SHA-256 040eebe747815557014c1dbf1d4265d204aaae35c110595f2a15b94ee7f68ca0