Spacetime-group visualizations

◦ 0 nontrivial lifts

A torus; translations only.

2222 2 nontrivial lifts

A sphere with four order-2 cone points.

  1. 22222222 g6

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    0.000
    Static perfect 2-colouring for group g6: asymmetric motifs carry phase colours for Conway type 2222/◦.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AHalf-turn rotation (180°)+1/2 period

    Relations G/Λ = ⟨A | A² = 1⟩

    Identifications

    • The Symmetries of Things · p. 141
    • g6
    • No. 2 p2
    • No. 1 p1
    • (21212121)
    • No. 4 P21Hall P 2yb
    • UCL diagram and tables

  2. 12122222 g7

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    0.000
    Static perfect 2-colouring for group g7: asymmetric motifs carry phase colours for Conway type 2222/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    A1/2-cell translation along b+1/2 period
    BHalf-turn rotation (180°)none

    Relations G/Λ = ⟨A, B | A² = B² = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 141
    • g7
    • No. 2 p2
    • No. 2 p2
    • (20202121)
    • No. 5 C2Hall C 2y
    • UCL diagram and tables

∗∗ 0 nontrivial lifts

Two mirror-boundary components.

×× 0 nontrivial lifts

Two crosscaps; glide reflections but no mirror boundary.

∗× 1 nontrivial lift

One mirror boundary and one crosscap.

  1. ∗2×2 g9

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    0.000
    Static perfect 2-colouring for group g9: asymmetric motifs carry phase colours for Conway type *×/◦.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection+1/2 period

    Relations G/Λ = ⟨A | A² = 1⟩

    Identifications

    • The Symmetries of Things · p. 141
    • g9
    • No. 5 c1m1 · cm
    • No. 1 p1
    • (∗:×)
    • No. 9 CcHall C -2yc
    • UCL diagram and tables

∗2222 5 nontrivial lifts

A mirror quadrilateral with four order-2 corners.

  1. ∗22222222 g55

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    0.000
    Static perfect 2-colouring for group g55: asymmetric motifs carry phase colours for Conway type *2222/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1+1/2 period
    BMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g55
    • No. 6 p2mm · pmm
    • No. 2 p2
    • (∗:2:2:2:2)
    • No. 27 Pcc2Hall P 2 -2c
    • UCL diagram and tables

  2. ∗12221222 g60

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    0.000
    Static perfect 2-colouring for group g60: asymmetric motifs carry phase colours for Conway type *2222/**.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1none
    BMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g60
    • No. 6 p2mm · pmm
    • No. 3 p1m1 · pm
    • (∗·2:2·2:2)
    • No. 26 Pmc21Hall P 2c -2
    • UCL diagram and tables

  3. ∗12121222 g64

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    0.000
    Static perfect 2-colouring for group g64: asymmetric motifs carry phase colours for Conway type *2222/*2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    A1/2-cell translation along b+1/2 period
    BMirror reflection in axis direction 1none
    CMirror reflection in axis direction 2none

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g64
    • No. 6 p2mm · pmm
    • No. 6 p2mm · pmm
    • (∗·2·2·2:2)
    • No. 38 Amm2Hall A 2 -2
    • UCL diagram and tables

  4. ∗12222222 g65

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    0.000
    Static perfect 2-colouring for group g65: asymmetric motifs carry phase colours for Conway type *2222/22*.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    A1/2-cell translation along b+1/2 period
    BMirror reflection in axis direction 1+1/2 period
    CMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g65
    • No. 6 p2mm · pmm
    • No. 7 p2mg · pmg
    • (∗·2:2:2:2)
    • No. 39 Aem2Hall A 2 -2b
    • UCL diagram and tables

  5. ∗12122222 g74

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    0.000
    Static perfect 2-colouring for group g74: asymmetric motifs carry phase colours for Conway type *2222/2*22.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    A1/2-cell translation along b+1/2 period
    B1/2-cell translation along a+1/2 period
    CMirror reflection in axis direction 1none
    DMirror reflection in axis direction 2none

    Relations G/Λ = ⟨A, B, C, D | A² = B² = C² = D² = 1; AB = BA, AC = CA, AD = DA, BC = CB, BD = DB, CD = DC⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g74
    • No. 6 p2mm · pmm
    • No. 9 c2mm · cmm
    • (∗·2·2:2:2)
    • No. 42 Fmm2Hall F 2 -2
    • UCL diagram and tables

22∗ 5 nontrivial lifts

Two order-2 cone points and one mirror-boundary component.

  1. 1212∗2 g57

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    0.000
    Static perfect 2-colouring for group g57: asymmetric motifs carry phase colours for Conway type 22*/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection+1/2 period
    BHalf-turn rotation (180°)none

    Relations G/Λ = ⟨A, B | A² = B² = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g57
    • No. 7 p2mg · pmg
    • No. 2 p2
    • (2020∗:)
    • No. 30 Pnc2Hall P 2 -2bc
    • UCL diagram and tables

  2. 2222∗2 g61

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    0.000
    Static perfect 2-colouring for group g61: asymmetric motifs carry phase colours for Conway type 22*/××.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection+1/2 period
    BHalf-turn rotation (180°)+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g61
    • No. 7 p2mg · pmg
    • No. 4 p1g1 · pg
    • (2121∗:)
    • No. 29 Pca21Hall P 2c -2ac
    • UCL diagram and tables

  3. 2222∗1 g62

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    0.000
    Static perfect 2-colouring for group g62: asymmetric motifs carry phase colours for Conway type 22*/**.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflectionnone
    BHalf-turn rotation (180°)+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g62
    • No. 7 p2mg · pmg
    • No. 3 p1m1 · pm
    • (2121∗·)
    • No. 31 Pmn21Hall P 2ac -2
    • UCL diagram and tables

  4. 1222∗1 g66

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    0.000
    Static perfect 2-colouring for group g66: asymmetric motifs carry phase colours for Conway type 22*/22*.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    A1/2-cell translation along b+1/2 period
    BMirror reflectionnone
    CHalf-turn rotation (180°)none

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g66
    • No. 7 p2mg · pmg
    • No. 7 p2mg · pmg
    • (2021∗·)
    • No. 40 Ama2Hall A 2 -2a
    • UCL diagram and tables

  5. 1222∗2 g67

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    0.000
    Static perfect 2-colouring for group g67: asymmetric motifs carry phase colours for Conway type 22*/22×.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    A1/2-cell translation along b+1/2 period
    BMirror reflection+1/2 period
    CHalf-turn rotation (180°)none

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g67
    • No. 7 p2mg · pmg
    • No. 8 p2gg · pgg
    • (2021∗:)
    • No. 41 Aea2Hall A 2 -2ab
    • UCL diagram and tables

22× 3 nontrivial lifts

Two order-2 cone points and one crosscap.

  1. 1212×2 g59

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    0.000
    Static perfect 2-colouring for group g59: asymmetric motifs carry phase colours for Conway type 22×/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AHalf-turn rotation (180°)none
    BGlide reflection+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 141
    • g59
    • No. 8 p2gg · pgg
    • No. 2 p2
    • (2020×1)
    • No. 34 Pnn2Hall P 2 -2n
    • UCL diagram and tables

  2. 2222×1 g63

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    0.000
    Static perfect 2-colouring for group g63: asymmetric motifs carry phase colours for Conway type 22×/××.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AHalf-turn rotation (180°)+1/2 period
    BGlide reflectionnone

    Relations G/Λ = ⟨A, B | A² = B² = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 141
    • g63
    • No. 8 p2gg · pgg
    • No. 4 p1g1 · pg
    • (2121×)
    • No. 33 Pna21Hall P 2c -2n
    • UCL diagram and tables

  3. 1222×4 g75

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    0.000
    Static perfect 4-colouring for group g75: asymmetric motifs carry phase colours for Conway type 22×⁴/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/4
    3. colour 2 · phase 1/2
    4. colour 3 · phase 3/4

    Presentation

    AHalf-turn rotation (180°)+1/2 period
    BGlide reflection+3/4 period

    Relations G/Λ = ⟨A, B | A² = B⁴ = (AB)⁴ = 1; AB² = B²A⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g75
    • No. 8 p2gg · pgg
    • No. 2 p2
    • (2021×)
    • No. 43 Fdd2Hall F 2 -2d
    • UCL diagram and tables

2∗22 5 nontrivial lifts

One order-2 cone point and a mirror boundary with two order-2 corners.

  1. 12∗2222 g69

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    0.000
    Static perfect 2-colouring for group g69: asymmetric motifs carry phase colours for Conway type 2*22/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    ATranslation by 1/2 a + 1/2 bnone
    BMirror reflection in axis direction 1+1/2 period
    CMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g69
    • No. 9 c2mm · cmm
    • No. 2 p2
    • (20∗:2:2)
    • No. 37 Ccc2Hall C 2 -2c
    • UCL diagram and tables

  2. 22∗1222 g70

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    0.000
    Static perfect 2-colouring for group g70: asymmetric motifs carry phase colours for Conway type 2*22/*×.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    ATranslation by 1/2 a + 1/2 bnone
    BMirror reflection in axis direction 1none
    CMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g70
    • No. 9 c2mm · cmm
    • No. 5 c1m1 · cm
    • (21∗·2:2)
    • No. 36 Cmc21Hall C 2c -2
    • UCL diagram and tables

  3. 22∗1212 g71

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    0.000
    Static perfect 2-colouring for group g71: asymmetric motifs carry phase colours for Conway type 2*22/*2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    ATranslation by 1/2 a + 1/2 b+1/2 period
    BMirror reflection in axis direction 1none
    CMirror reflection in axis direction 2none

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g71
    • No. 9 c2mm · cmm
    • No. 6 p2mm · pmm
    • (21∗·2·2)
    • No. 44 Imm2Hall I 2 -2
    • UCL diagram and tables

  4. 22∗2222 g72

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    0.000
    Static perfect 2-colouring for group g72: asymmetric motifs carry phase colours for Conway type 2*22/22×.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    ATranslation by 1/2 a + 1/2 b+1/2 period
    BMirror reflection in axis direction 1+1/2 period
    CMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g72
    • No. 9 c2mm · cmm
    • No. 8 p2gg · pgg
    • (21∗:2:2)
    • No. 45 Iba2Hall I 2 -2c
    • UCL diagram and tables

  5. 12∗1222 g73

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    0.000
    Static perfect 2-colouring for group g73: asymmetric motifs carry phase colours for Conway type 2*22/22*.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    ATranslation by 1/2 a + 1/2 b+1/2 period
    BMirror reflection in axis direction 1+1/2 period
    CMirror reflection in axis direction 2none

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = 1; AB = BA, AC = CA, BC = CB⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g73
    • No. 9 c2mm · cmm
    • No. 7 p2mg · pmg
    • (20∗·2:2)
    • No. 46 Ima2Hall I 2 -2a
    • UCL diagram and tables

442 5 nontrivial lifts

A sphere with cone points of orders 4, 4, and 2.

  1. 242412 g95

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    0.000
    Static perfect 2-colouring for group g95: asymmetric motifs carry phase colours for Conway type 442/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    A1/4-turn rotation (90°)+1/2 period

    Relations G/Λ = ⟨A | A⁴ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g95
    • No. 10 p4
    • No. 2 p2
    • (424220)
    • No. 77 P42Hall P 4c
    • UCL diagram and tables

  2. 444422 g96

    414121
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    0.000
    Static perfect 4-colouring for group g96: asymmetric motifs carry phase colours for Conway type 442⁴/◦.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/4
    3. colour 2 · phase 1/2
    4. colour 3 · phase 3/4

    Presentation

    A1/4-turn rotation (90°)+1/4 period

    Relations G/Λ = ⟨A | A⁴ = 1⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g96
    • No. 10 p4
    • No. 1 p1
    • (414121)Two orientations share this fibrifold name; the space-group name below selects this handed form.
    • No. 76 P41Hall P 4w
    • UCL diagram and tables
    • g97

  3. 444422 g97

    434321
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    0.000
    Static perfect 4-colouring for group g97: asymmetric motifs carry phase colours for Conway type 442⁴/◦.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/4
    3. colour 2 · phase 1/2
    4. colour 3 · phase 3/4

    Presentation

    A1/4-turn rotation (90°)+3/4 period

    Relations G/Λ = ⟨A | A⁴ = 1⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g97
    • No. 10 p4
    • No. 1 p1
    • (414121)Two orientations share this fibrifold name; the space-group name below selects this handed form.
    • No. 78 P43Hall P 4cw
    • UCL diagram and tables
    • g96

  4. 142422 g98

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    0.000
    Static perfect 2-colouring for group g98: asymmetric motifs carry phase colours for Conway type 442/442.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    ATranslation by 1/2 a + 1/2 b+1/2 period
    B1/4-turn rotation (90°)none

    Relations G/Λ = ⟨A, B | A² = B⁴ = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g98
    • No. 10 p4
    • No. 10 p4
    • (424021)
    • No. 79 I4Hall I 4
    • UCL diagram and tables

  5. 444412 g99

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    0.000
    Static perfect 4-colouring for group g99: asymmetric motifs carry phase colours for Conway type 442⁴/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/4
    3. colour 2 · phase 1/2
    4. colour 3 · phase 3/4

    Presentation

    ATranslation by 1/2 a + 1/2 b+1/2 period
    B1/4-turn rotation (90°)+3/4 period

    Relations G/Λ = ⟨A, B | A² = B⁴ = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g99
    • No. 10 p4
    • No. 2 p2
    • (434120)
    • No. 80 I41Hall I 4bw
    • UCL diagram and tables

∗442 5 nontrivial lifts

A mirror triangle with corner orders 4, 4, and 2.

  1. ∗142412 g129

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    0.000
    Static perfect 2-colouring for group g129: asymmetric motifs carry phase colours for Conway type *442/*2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1none
    BMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = (AB)⁴ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g129
    • No. 11 p4mm · p4m
    • No. 6 p2mm · pmm
    • (∗·4:4·2)
    • No. 105 P42mcHall P 4c -2
    • UCL diagram and tables

  2. ∗241422 g130

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    0.000
    Static perfect 2-colouring for group g130: asymmetric motifs carry phase colours for Conway type *442/2*22.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1+1/2 period
    BMirror reflection in axis direction 2none

    Relations G/Λ = ⟨A, B | A² = B² = (AB)⁴ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g130
    • No. 11 p4mm · p4m
    • No. 9 c2mm · cmm
    • (∗:4·4:2)
    • No. 101 P42cmHall P 4c -2c
    • UCL diagram and tables

  3. ∗242422 g131

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    0.000
    Static perfect 2-colouring for group g131: asymmetric motifs carry phase colours for Conway type *442/442.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1+1/2 period
    BMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = (AB)⁴ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g131
    • No. 11 p4mm · p4m
    • No. 10 p4
    • (∗:4:4:2)
    • No. 103 P4ccHall P 4 -2c
    • UCL diagram and tables

  4. ∗141422 g136

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    0.000
    Static perfect 2-colouring for group g136: asymmetric motifs carry phase colours for Conway type *442/*442.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    ATranslation by 1/2 a + 1/2 b+1/2 period
    BMirror reflection in axis direction 1none
    CMirror reflection in axis direction 2none

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = (BC)⁴ = 1; AB = BA, AC = CA⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g136
    • No. 11 p4mm · p4m
    • No. 11 p4mm · p4m
    • (∗·4·4:2)
    • No. 107 I4mmHall I 4 -2
    • UCL diagram and tables

  5. ∗142422 g138

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    0.000
    Static perfect 2-colouring for group g138: asymmetric motifs carry phase colours for Conway type *442/4*2.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    ATranslation by 1/2 a + 1/2 b+1/2 period
    BMirror reflection in axis direction 1+1/2 period
    CMirror reflection in axis direction 2none

    Relations G/Λ = ⟨A, B, C | A² = B² = C² = (BC)⁴ = 1; AB = BA, AC = CA⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g138
    • No. 11 p4mm · p4m
    • No. 12 p4gm · p4g
    • (∗·4:4:2)
    • No. 108 I4cmHall I 4 -2c
    • UCL diagram and tables

4∗2 5 nontrivial lifts

One order-4 cone point and a mirror boundary with one order-2 corner.

  1. 24∗22 g133

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    0.000
    Static perfect 2-colouring for group g133: asymmetric motifs carry phase colours for Conway type 4*2/22×.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection+1/2 period
    B1/4-turn rotation (90°)+1/2 period

    Relations G/Λ = ⟨A, B | A² = B⁴ = (AB)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g133
    • No. 12 p4gm · p4g
    • No. 8 p2gg · pgg
    • (42∗:2)
    • No. 106 P42bcHall P 4c -2ab
    • UCL diagram and tables

  2. 24∗12 g134

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    0.000
    Static perfect 2-colouring for group g134: asymmetric motifs carry phase colours for Conway type 4*2/2*22.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflectionnone
    B1/4-turn rotation (90°)+1/2 period

    Relations G/Λ = ⟨A, B | A² = B⁴ = (AB)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g134
    • No. 12 p4gm · p4g
    • No. 9 c2mm · cmm
    • (42∗·2)
    • No. 102 P42nmHall P 4n -2n
    • UCL diagram and tables

  3. 14∗22 g135

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    0.000
    Static perfect 2-colouring for group g135: asymmetric motifs carry phase colours for Conway type 4*2/442.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection+1/2 period
    B1/4-turn rotation (90°)none

    Relations G/Λ = ⟨A, B | A² = B⁴ = (AB)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g135
    • No. 12 p4gm · p4g
    • No. 10 p4
    • (40∗:2)
    • No. 104 P4ncHall P 4 -2n
    • UCL diagram and tables

  4. 44∗12 g137

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    0.000
    Static perfect 4-colouring for group g137: asymmetric motifs carry phase colours for Conway type 4*2⁴/*2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/4
    3. colour 2 · phase 1/2
    4. colour 3 · phase 3/4

    Presentation

    AMirror reflectionnone
    B1/4-turn rotation (90°)+3/4 period

    Relations G/Λ = ⟨A, B | A² = B⁴ = (AB)⁴ = 1; AB² = B²A⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g137
    • No. 12 p4gm · p4g
    • No. 6 p2mm · pmm
    • (41∗·2)
    • No. 109 I41mdHall I 4bw -2
    • UCL diagram and tables

  5. 44∗22 g139

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    0.000
    Static perfect 4-colouring for group g139: asymmetric motifs carry phase colours for Conway type 4*2⁴/22×.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/4
    3. colour 2 · phase 1/2
    4. colour 3 · phase 3/4

    Presentation

    AMirror reflection+1/2 period
    B1/4-turn rotation (90°)+1/4 period

    Relations G/Λ = ⟨A, B | A² = B⁴ = (AB)⁴ = 1; AB² = B²A⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g139
    • No. 12 p4gm · p4g
    • No. 8 p2gg · pgg
    • (41∗:2)
    • No. 110 I41cdHall I 4bw -2c
    • UCL diagram and tables

333 3 nontrivial lifts

A sphere with three order-3 cone points.

  1. 333333 g225

    323232
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    0.000
    Static perfect 3-colouring for group g225: asymmetric motifs carry phase colours for Conway type 333³/◦.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/3
    3. colour 2 · phase 2/3

    Presentation

    A1/3-turn rotation (120°)+1/3 period

    Relations G/Λ = ⟨A | A³ = 1⟩

    Identifications

    • The Symmetries of Things · p. 156
    • g225
    • No. 13 p3
    • No. 1 p1
    • (313131)Two orientations share this fibrifold name; the space-group name below selects this handed form.
    • No. 145 P32Hall P 32
    • UCL diagram and tables
    • g226

  2. 333333 g226

    313131
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    0.000
    Static perfect 3-colouring for group g226: asymmetric motifs carry phase colours for Conway type 333³/◦.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/3
    3. colour 2 · phase 2/3

    Presentation

    A1/3-turn rotation (120°)+2/3 period

    Relations G/Λ = ⟨A | A³ = 1⟩

    Identifications

    • The Symmetries of Things · p. 156
    • g226
    • No. 13 p3
    • No. 1 p1
    • (313131)Two orientations share this fibrifold name; the space-group name below selects this handed form.
    • No. 144 P31Hall P 31
    • UCL diagram and tables
    • g225

  3. 333313 g227

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    0.000
    Static perfect 3-colouring for group g227: asymmetric motifs carry phase colours for Conway type 333³/333.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/3
    3. colour 2 · phase 2/3

    Presentation

    ATranslation by 1/3 a − 1/3 b+2/3 period
    B1/3-turn rotation (120°)none

    Relations G/Λ = ⟨A, B | A³ = B³ = 1; AB = BA⟩

    Identifications

    • The Symmetries of Things · p. 156
    • g227
    • No. 13 p3
    • No. 13 p3
    • (303132)
    • No. 146 R3Hall R 3
    • UCL diagram and tables

∗333 1 nontrivial lift

A mirror triangle with three order-3 corners.

  1. ∗232323 g231

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    0.000
    Static perfect 2-colouring for group g231: asymmetric motifs carry phase colours for Conway type *333/333.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1+1/2 period
    BMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = (AB)³ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g231
    • No. 14 p3m1
    • No. 13 p3
    • (∗:3:3:3)
    • No. 158 P3c1Hall P 3 -2"c
    • UCL diagram and tables

3∗3 3 nontrivial lifts

One order-3 cone point and a mirror boundary with one order-3 corner.

  1. 13∗23 g233

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    0.000
    Static perfect 2-colouring for group g233: asymmetric motifs carry phase colours for Conway type 3*3/333.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1+1/2 period
    BMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = (AB)³ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g233
    • No. 15 p31m
    • No. 13 p3
    • (30∗:3)
    • No. 159 P31cHall P 3 -2c
    • UCL diagram and tables

  2. 33∗13 g234

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    0.000
    Static perfect 3-colouring for group g234: asymmetric motifs carry phase colours for Conway type 3*3³/*333.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/3
    3. colour 2 · phase 2/3

    Presentation

    AMirror reflectionnone
    B1/3-turn rotation (120°)+2/3 period

    Relations G/Λ = ⟨A, B | A² = B³ = 1; B(ABA) = (ABA)B⟩

    Identifications

    • The Symmetries of Things · p. 164
    • g234
    • No. 15 p31m
    • No. 14 p3m1
    • (31∗·3)
    • No. 160 R3mHall R 3 -2"
    • UCL diagram and tables

  3. 33∗23 g235

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    0.000
    Static perfect 6-colouring for group g235: asymmetric motifs carry phase colours for Conway type 3*3⁶/333.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/6
    3. colour 2 · phase 1/3
    4. colour 3 · phase 1/2
    5. colour 4 · phase 2/3
    6. colour 5 · phase 5/6

    Presentation

    AMirror reflection+1/2 period
    B1/3-turn rotation (120°)+2/3 period

    Relations G/Λ = ⟨A, B | A² = B³ = 1; B(ABA) = (ABA)B⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g235
    • No. 15 p31m
    • No. 13 p3
    • (31∗:3)
    • No. 161 R3cHall R 3 -2"c
    • UCL diagram and tables

632 5 nontrivial lifts

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

  1. 363312 g244

    62322
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    0.000
    Static perfect 3-colouring for group g244: asymmetric motifs carry phase colours for Conway type 632³/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/3
    3. colour 2 · phase 2/3

    Presentation

    A1/6-turn rotation (60°)+2/3 period

    Relations G/Λ = ⟨A | A⁶ = 1⟩

    Identifications

    • The Symmetries of Things · p. 164
    • g244
    • No. 16 p6
    • No. 2 p2
    • (623220)Two orientations share this fibrifold name; the space-group name below selects this handed form.
    • No. 171 P62Hall P 62
    • UCL diagram and tables
    • g245

  2. 363312 g245

    64312
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    0.000
    Static perfect 3-colouring for group g245: asymmetric motifs carry phase colours for Conway type 632³/2222.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/3
    3. colour 2 · phase 2/3

    Presentation

    A1/6-turn rotation (60°)+1/3 period

    Relations G/Λ = ⟨A | A⁶ = 1⟩

    Identifications

    • The Symmetries of Things · p. 164
    • g245
    • No. 16 p6
    • No. 2 p2
    • (623220)Two orientations share this fibrifold name; the space-group name below selects this handed form.
    • No. 172 P64Hall P 64
    • UCL diagram and tables
    • g244

  3. 261322 g246

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    0.000
    Static perfect 2-colouring for group g246: asymmetric motifs carry phase colours for Conway type 632/333.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    A1/6-turn rotation (60°)+1/2 period

    Relations G/Λ = ⟨A | A⁶ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g246
    • No. 16 p6
    • No. 13 p3
    • (633021)
    • No. 173 P63Hall P 6c
    • UCL diagram and tables

  4. 663322 g247

    653221
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    0.000
    Static perfect 6-colouring for group g247: asymmetric motifs carry phase colours for Conway type 632⁶/◦.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/6
    3. colour 2 · phase 1/3
    4. colour 3 · phase 1/2
    5. colour 4 · phase 2/3
    6. colour 5 · phase 5/6

    Presentation

    A1/6-turn rotation (60°)+1/6 period

    Relations G/Λ = ⟨A | A⁶ = 1⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g247
    • No. 16 p6
    • No. 1 p1
    • (613121)Two orientations share this fibrifold name; the space-group name below selects this handed form.
    • No. 170 P65Hall P 65
    • UCL diagram and tables
    • g248

  5. 663322 g248

    613121
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    0.000
    Static perfect 6-colouring for group g248: asymmetric motifs carry phase colours for Conway type 632⁶/◦.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/6
    3. colour 2 · phase 1/3
    4. colour 3 · phase 1/2
    5. colour 4 · phase 2/3
    6. colour 5 · phase 5/6

    Presentation

    A1/6-turn rotation (60°)+5/6 period

    Relations G/Λ = ⟨A | A⁶ = 1⟩

    Identifications

    • The Symmetries of Things · p. 155
    • g248
    • No. 16 p6
    • No. 1 p1
    • (613121)Two orientations share this fibrifold name; the space-group name below selects this handed form.
    • No. 169 P61Hall P 61
    • UCL diagram and tables
    • g247

∗632 3 nontrivial lifts

A mirror triangle with corner orders 6, 3, and 2.

  1. ∗162322 g269

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    0.000
    Static perfect 2-colouring for group g269: asymmetric motifs carry phase colours for Conway type *632/3*3.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1none
    BMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = (AB)⁶ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g269
    • No. 17 p6mm · p6m
    • No. 15 p31m
    • (∗·6:3:2)
    • No. 185 P63cmHall P 6c -2
    • UCL diagram and tables

  2. ∗262322 g270

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    0.000
    Static perfect 2-colouring for group g270: asymmetric motifs carry phase colours for Conway type *632/632.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1+1/2 period
    BMirror reflection in axis direction 2+1/2 period

    Relations G/Λ = ⟨A, B | A² = B² = (AB)⁶ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g270
    • No. 17 p6mm · p6m
    • No. 16 p6
    • (∗:6:3:2)
    • No. 184 P6ccHall P 6 -2c
    • UCL diagram and tables

  3. ∗261312 g271

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    0.000
    Static perfect 2-colouring for group g271: asymmetric motifs carry phase colours for Conway type *632/*333.
    1. colour 0 · phase 0
    2. colour 1 · phase 1/2

    Presentation

    AMirror reflection in axis direction 1+1/2 period
    BMirror reflection in axis direction 2none

    Relations G/Λ = ⟨A, B | A² = B² = (AB)⁶ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • g271
    • No. 17 p6mm · p6m
    • No. 14 p3m1
    • (∗:6·3·2)
    • No. 186 P63mcHall P 6c -2c
    • UCL diagram and tables

Data

68-record JSON · generator · book-excerpt map · errata · SHA-256 040eebe747815557014c1dbf1d4265d204aaae35c110595f2a15b94ee7f68ca0