Spacetime-group visualizations

Diagram key

Crystallographic generator symbols: visual index

Projected along the clock axis, these are the actual symbols used on the static 2D diagrams. Open arrows mark primitive translations in the picture; the crystallographic filled core shows the 2-, 3-, 4-, or 6-fold spatial order. For a forward skip τ, the screw subscript is m ≡ nτ (mod n) for a positive 1/n-turn and m ≡ −nτ (mod n) for the opposite turn, with 0 ≤ m < n. Mirrored blades record spatial turn sense—not backward time.

Translations in the projection plane

  • Primitive translationOpen arrow; its direction is the named in-plane translation X, Y, or α.

Rotation and screw axes (along the clock axis)

  • 2-fold rotation axisFilled lens; half-turn with no clock-axis rise.
  • 21 screw axisTwo-arm lens; half-turn with a +1/2-period rise.
  • 3-fold rotation axisFilled triangle; one-third turn with no clock-axis rise.
  • 31 screw axisThree-stroke triangle; positive one-third turn with a +1/3-period rise.
  • 32 screw axisOpposite three-stroke triangle; positive one-third turn with a +2/3-period rise.
  • 4-fold rotation axisFilled diamond; quarter-turn with no clock-axis rise.
  • 41 screw axisFour-arm diamond; positive quarter-turn with a +1/4-period rise.
  • 42 screw axisTwo-arm diamond; quarter-turn with a +1/2-period rise.
  • 43 screw axisOpposite four-arm diamond; positive quarter-turn with a +3/4-period rise.
  • 6-fold rotation axisFilled hexagon; one-sixth turn with no clock-axis rise.
  • 61 screw axisSix-arm hexagon; positive one-sixth turn with a +1/6-period rise.
  • 62 screw axisThree-arm hexagon; positive one-sixth turn with a +1/3-period rise.
  • 63 screw axisTwo-arm hexagon; one-sixth turn with a +1/2-period rise.
  • 64 screw axisOpposite three-arm hexagon; positive one-sixth turn with a +2/3-period rise.
  • 65 screw axisOpposite six-arm hexagon; positive one-sixth turn with a +5/6-period rise.

Mirror and glide planes (seen edge-on)

  • Mirror plane (m)Solid line; reflection with no glide component.
  • In-plane axial glideDashed line; reflection plus a half-cell step in the picture.
  • Clock-axis glide (c)Dotted line; reflection plus a half-period step perpendicular to the picture.
  • Diagonal n-glideDash–dot line; half-steps in the picture and along the clock axis.
  • Quarter-diagonal d-glideArrowed dash–dot line; arrows select the sign of the quarter-period component.

The burgundy α, β, γ, … or P, Q, … beside a diagram symbol identifies its Generator row. The Color and Time columns state the same forward clock action directly.

Notation: International Tables for Crystallography projection symbols · UCL P 3 1 c example.

◦ 1 forward group

A torus; translations only.

  1. ◦

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g1: asymmetric motifs carry phase colours for Conway type ◦.
    Space–time lift diagram for g1 and polar space group No. 1 P1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Gilmarite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · GilmariteNo. 1 P1 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Xtranslationnonenone
    Ytranslationnonenone

    Relations Γ = ⟨X, Y | XY = YX⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(◦)P1No. 1C11P 1g1
    Extra links

2222 3 forward groups

A sphere with four order-2 cone points.

  1. 22222222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g6: asymmetric motifs carry phase colours for Conway type 2222/◦.
    Space–time lift diagram for g6 and polar space group No. 4 P2_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Alloclasite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · AlloclasiteNo. 4 P21 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    δhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period

    Relations Γ = ⟨α, β, γ, δ | α² = β² = γ² = δ² = αβγδ = 1⟩

    Identifications

    • The Symmetries of Things · p. 141
    • Other names(21212121)P21P 1 21 1No. 4C22P 2ybg6
    Extra links

  2. 12122222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g7: asymmetric motifs carry phase colours for Conway type 2222/2222.
    Space–time lift diagram for g7 and polar space group No. 5 C2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Miargyrite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · MiargyriteNo. 5 C2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Same-space-group substitute. The CrystalSymmetry No. 5 page names PANWAC but links no public interactive model; Miargyrite is an explicitly labelled replacement in the same C2 space group.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-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
    δhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period

    Relations Γ = ⟨α, β, γ, δ | α² = β² = γ² = δ² = αβγδ = 1⟩

    Identifications

    • The Symmetries of Things · p. 141
    • Other names(20202121)C2C 1 2 1No. 5C23C 2yg7
    Extra links

  3. 2222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g5: asymmetric motifs carry phase colours for Conway type 2222.
    Space–time lift diagram for g5 and polar space group No. 3 P2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Thomasclarkite-(Y).
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Thomasclarkite-(Y)No. 3 P2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-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
    δhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone

    Relations Γ = ⟨α, β, γ, δ | α² = β² = γ² = δ² = αβγδ = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(20202020)P2P 1 2 1No. 3C21P 2yg5
    Extra links

∗∗ 1 forward group

Two mirror-boundary components.

  1. ∗∗

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g10: asymmetric motifs carry phase colours for Conway type **.
    Space–time lift diagram for g10 and polar space group No. 6 Pm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Clinobarylite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · ClinobaryliteNo. 6 Pm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αtranslationnonenone
    Pmirror reflectionnonenone
    Qparallel mirror reflectionnonenone

    Relations Γ = ⟨α, P, Q | P² = Q² = 1; αP = Pα; αQ = Qα⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(∗·∗·)PmP 1 m 1No. 6Cs1P -2yg10
    Extra links

×× 1 forward group

Two crosscaps; glide reflections but no mirror boundary.

  1. ××

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g11: asymmetric motifs carry phase colours for Conway type ××.
    Space–time lift diagram for g11 and polar space group No. 7 Pc; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Tashelgite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · TashelgiteNo. 7 Pc CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Yglide reflectionnonenone
    Zglide reflectionnonenone

    Relations Γ = ⟨Y, Z | Y²Z² = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(××0)PcP 1 c 1No. 7Cs2P -2ycg11
    Extra links

∗× 2 forward groups

One mirror boundary and one crosscap.

  1. ∗2×2

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g9: asymmetric motifs carry phase colours for Conway type *×/◦.
    Space–time lift diagram for g9 and polar space group No. 9 Cc; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Fe(Mo2S4).
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Fe(Mo2S4)No. 9 Cc CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionC2+1/2 period
    Zglide reflectionC2+1/2 period

    Relations Γ = ⟨P, Z | P² = PZ²PZ⁻² = 1⟩

    Identifications

    • The Symmetries of Things · p. 141
    • Other names(∗:×)CcC 1 c 1No. 9Cs4C -2ycg9
    Extra links

  2. ∗×

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g8: asymmetric motifs carry phase colours for Conway type *×.
    Space–time lift diagram for g8 and polar space group No. 8 Cm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Gerstleyite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · GerstleyiteNo. 8 Cm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Zglide reflectionnonenone

    Relations Γ = ⟨P, Z | P² = PZ²PZ⁻² = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(∗·×)CmC 1 m 1No. 8Cs3C -2yg8
    Extra links

∗2222 6 forward groups

A mirror quadrilateral with four order-2 corners.

  1. ∗22222222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g55: asymmetric motifs carry phase colours for Conway type *2222/2222.
    Space–time lift diagram for g55 and polar space group No. 27 Pcc2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine 2,5- Bis(5-bromo-2-thienyl)thiophene.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · 2,5- Bis(5-bromo-2-thienyl)thiopheneNo. 27 Pcc2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionC2+1/2 period
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionC2+1/2 period
    Smirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R, S | P² = Q² = R² = S² = (PQ)² = (QR)² = (RS)² = (SP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗:2:2:2:2)Pcc2No. 27C2v3P 2 -2cg55
    Extra links

  2. ∗12221222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g60: asymmetric motifs carry phase colours for Conway type *2222/**.
    Space–time lift diagram for g60 and polar space group No. 26 Pmc2_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Carbocernaite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · CarbocernaiteNo. 26 Pmc21 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionnonenone
    Smirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R, S | P² = Q² = R² = S² = (PQ)² = (QR)² = (RS)² = (SP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗·2:2·2:2)Pmc21No. 26C2v2P 2c -2g60
    Extra links

  3. ∗12121222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g64: asymmetric motifs carry phase colours for Conway type *2222/*2222.
    Space–time lift diagram for g64 and polar space group No. 38 Amm2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Shortite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · ShortiteNo. 38 Amm2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone
    Rmirror reflectionnonenone
    Smirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R, S | P² = Q² = R² = S² = (PQ)² = (QR)² = (RS)² = (SP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗·2·2·2:2)Amm2No. 38C2v14A 2 -2g64
    Extra links

  4. ∗12222222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g65: asymmetric motifs carry phase colours for Conway type *2222/22*.
    Space–time lift diagram for g65 and polar space group No. 39 Aem2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine 2,5-bis(4- Br-benzylidene)c-pentanone.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · 2,5-bis(4- Br-benzylidene)c-pentanoneNo. 39 Aem2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionC2+1/2 period
    Smirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R, S | P² = Q² = R² = S² = (PQ)² = (QR)² = (RS)² = (SP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗·2:2:2:2)Aem2No. 39C2v15A 2 -2bg65
    Extra links

  5. ∗12122222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g74: asymmetric motifs carry phase colours for Conway type *2222/2*22.
    Space–time lift diagram for g74 and polar space group No. 42 Fmm2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine CH2I2.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · CH2I2No. 42 Fmm2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone
    Rmirror reflectionC2+1/2 period
    Smirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R, S | P² = Q² = R² = S² = (PQ)² = (QR)² = (RS)² = (SP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗·2·2:2:2)Fmm2No. 42C2v18F 2 -2g74
    Extra links

  6. ∗2222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g54: asymmetric motifs carry phase colours for Conway type *2222.
    Space–time lift diagram for g54 and polar space group No. 25 Pmm2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine GaAs.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · GaAsNo. 25 Pmm2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone
    Rmirror reflectionnonenone
    Smirror reflectionnonenone

    Relations Γ = ⟨P, Q, R, S | P² = Q² = R² = S² = (PQ)² = (QR)² = (RS)² = (SP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(∗·2·2·2·2)Pmm2No. 25C2v1P 2 -2g54
    Extra links

22∗ 6 forward groups

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

  1. 1212∗2

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g57: asymmetric motifs carry phase colours for Conway type 22*/2222.
    Space–time lift diagram for g57 and polar space group No. 30 Pnc2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Terskite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · TerskiteNo. 30 Pnc2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Pmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, β, P | α² = β² = P² = 1; αβP = Pαβ⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(2020∗:)Pnc2No. 30C2v6P 2 -2bcg57
    Extra links

  2. 2222∗2

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g61: asymmetric motifs carry phase colours for Conway type 22*/××.
    Space–time lift diagram for g61 and polar space group No. 29 Pca2_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Cobaltite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · CobaltiteNo. 29 Pca21 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, β, P | α² = β² = P² = 1; αβP = Pαβ⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(2121∗:)Pca21No. 29C2v5P 2c -2acg61
    Extra links

  3. 2222∗1

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g62: asymmetric motifs carry phase colours for Conway type 22*/**.
    Space–time lift diagram for g62 and polar space group No. 31 Pmn2_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Enargite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · EnargiteNo. 31 Pmn21 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionnonenone

    Relations Γ = ⟨α, β, P | α² = β² = P² = 1; αβP = Pαβ⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(2121∗·)Pmn21No. 31C2v7P 2ac -2g62
    Extra links

  4. 1222∗1

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g66: asymmetric motifs carry phase colours for Conway type 22*/22*.
    Space–time lift diagram for g66 and polar space group No. 40 Ama2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine LiBH4 at 2 4 GPa.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · LiBH4 at 2 4 GPaNo. 40 Ama2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionnonenone

    Relations Γ = ⟨α, β, P | α² = β² = P² = 1; αβP = Pαβ⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(2021∗·)Ama2No. 40C2v16A 2 -2ag66
    Extra links

  5. 1222∗2

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g67: asymmetric motifs carry phase colours for Conway type 22*/22×.
    Space–time lift diagram for g67 and polar space group No. 41 Aea2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Si2CN4.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Si2CN4No. 41 Aea2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, β, P | α² = β² = P² = 1; αβP = Pαβ⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(2021∗:)Aea2No. 41C2v17A 2 -2abg67
    Extra links

  6. 22∗

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g56: asymmetric motifs carry phase colours for Conway type 22*.
    Space–time lift diagram for g56 and polar space group No. 28 Pma2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Krennerite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · KrenneriteNo. 28 Pma2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Pmirror reflectionnonenone

    Relations Γ = ⟨α, β, P | α² = β² = P² = 1; αβP = Pαβ⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(2020∗·)Pma2No. 28C2v4P 2 -2ag56
    Extra links

22× 4 forward groups

Two order-2 cone points and one crosscap.

  1. 1212×2

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g59: asymmetric motifs carry phase colours for Conway type 22×/2222.
    Space–time lift diagram for g59 and polar space group No. 34 Pnn2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Li2TiTeO6.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Li2TiTeO6No. 34 Pnn2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Zglide reflectionC2+1/2 period

    Relations Γ = ⟨α, β, Z | α² = β² = αβZ² = 1⟩

    Identifications

    • The Symmetries of Things · p. 141
    • Other names(2020×1)Pnn2No. 34C2v10P 2 -2ng59
    Extra links

  2. 2222×1

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g63: asymmetric motifs carry phase colours for Conway type 22×/××.
    Space–time lift diagram for g63 and polar space group No. 33 Pna2_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Wakabayashilite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · WakabayashiliteNo. 33 Pna21 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Zglide reflectionnonenone

    Relations Γ = ⟨α, β, Z | α² = β² = αβZ² = 1⟩

    Identifications

    • The Symmetries of Things · p. 141
    • Other names(2121×)Pna21No. 33C2v9P 2c -2ng63
    Extra links

  3. 1222×4

    Derived C4 short form — no literal book row

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 4-colouring for group g75: asymmetric motifs carry phase colours for Conway type 22×⁴/2222.
    Space–time lift diagram for g75 and polar space group No. 43 Fdd2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Edenharterite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · EdenharteriteNo. 43 Fdd2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C42+1/2 period
    Zglide reflectionC4+1/4 period

    Relations Γ = ⟨α, β, Z | α² = β² = αβZ² = 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
    • β (AB)(CD)
    • Z (ACBD)

    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 1212×2 · p. 141C2 layer 12122222 · p. 141
    • Other names(2021×)Fdd2No. 43C2v19F 2 -2dg75
    Extra links

  4. 22×

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g58: asymmetric motifs carry phase colours for Conway type 22×.
    Space–time lift diagram for g58 and polar space group No. 32 Pba2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Minyulite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · MinyuliteNo. 32 Pba2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Zglide reflectionnonenone

    Relations Γ = ⟨α, β, Z | α² = β² = αβZ² = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(2020×0)Pba2No. 32C2v8P 2 -2abg58
    Extra links

2∗22 6 forward groups

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

  1. 12∗2222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g69: asymmetric motifs carry phase colours for Conway type 2*22/2222.
    Space–time lift diagram for g69 and polar space group No. 37 Ccc2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine ZP-4.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · ZP-4No. 37 Ccc2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Pmirror reflectionC2+1/2 period
    Qmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, P, Q | α² = P² = Q² = (PQ)² = (QαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(20∗:2:2)Ccc2No. 37C2v13C 2 -2cg69
    Extra links

  2. 22∗1222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g70: asymmetric motifs carry phase colours for Conway type 2*22/*×.
    Space–time lift diagram for g70 and polar space group No. 36 Cmc2_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Spertiniite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · SpertiniiteNo. 36 Cmc21 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionnonenone
    Qmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, P, Q | α² = P² = Q² = (PQ)² = (QαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(21∗·2:2)Cmc21No. 36C2v12C 2c -2g70
    Extra links

  3. 22∗1212

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g71: asymmetric motifs carry phase colours for Conway type 2*22/*2222.
    Space–time lift diagram for g71 and polar space group No. 44 Imm2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine AgNO2.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · AgNO2No. 44 Imm2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone

    Relations Γ = ⟨α, P, Q | α² = P² = Q² = (PQ)² = (QαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(21∗·2·2)Imm2No. 44C2v20I 2 -2g71
    Extra links

  4. 22∗2222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g72: asymmetric motifs carry phase colours for Conway type 2*22/22×.
    Space–time lift diagram for g72 and polar space group No. 45 Iba2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Banalsite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · BanalsiteNo. 45 Iba2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionC2+1/2 period
    Qmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, P, Q | α² = P² = Q² = (PQ)² = (QαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(21∗:2:2)Iba2No. 45C2v21I 2 -2cg72
    Extra links

  5. 12∗1222

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g73: asymmetric motifs carry phase colours for Conway type 2*22/22*.
    Space–time lift diagram for g73 and polar space group No. 46 Ima2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Batisite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · BatisiteNo. 46 Ima2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Pmirror reflectionnonenone
    Qmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, P, Q | α² = P² = Q² = (PQ)² = (QαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(20∗·2:2)Ima2No. 46C2v22I 2 -2ag73
    Extra links

  6. 2∗22

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g68: asymmetric motifs carry phase colours for Conway type 2*22.
    Space–time lift diagram for g68 and polar space group No. 35 Cmm2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine KNbW2O9.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · KNbW2O9No. 35 Cmm2 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone

    Relations Γ = ⟨α, P, Q | α² = P² = Q² = (PQ)² = (QαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(20∗·2·2)Cmm2No. 35C2v11C 2 -2g68
    Extra links

442 6 forward groups

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

  1. 242412

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g95: asymmetric motifs carry phase colours for Conway type 442/2222.
    Space–time lift diagram for g95 and polar space group No. 77 P4_2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Pinnoite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · PinnoiteNo. 77 P42 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

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

    Relations Γ = ⟨α, β, γ | α⁴ = β⁴ = γ² = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(424220)P42No. 77C43P 4cg95
    Extra links

  2. 444422

    Derived C4 short form — no literal book row

    414121
    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 4-colouring for group g96: asymmetric motifs carry phase colours for Conway type 442⁴/◦.
    Space–time lift diagram for g96 and polar space group No. 76 P4_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Percleveite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · PercleveiteNo. 76 P41 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C43+3/4 period
    βquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C43+3/4 period
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C42+1/2 period

    Relations Γ = ⟨α, β, γ | α⁴ = β⁴ = 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 (ABDC)
    • β (ABDC)
    • γ (AD)(BC)

    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 242412 · p. 140C2 layer 22222222 · p. 141
    • Other names(414121)P41No. 76C42P 4wg97g96
    Extra links

  3. 444422

    Derived C4 short form — no literal book row

    434321
    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 4-colouring for group g97: asymmetric motifs carry phase colours for Conway type 442⁴/◦.
    Space–time lift diagram for g97 and polar space group No. 78 P4_3; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Sr2As2O7.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Sr2As2O7No. 78 P43 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C4+1/4 period
    βquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C4+1/4 period
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C42+1/2 period

    Relations Γ = ⟨α, β, γ | α⁴ = β⁴ = 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 (ABDC)
    • β (ABDC)
    • γ (AD)(BC)

    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 242412 · p. 140C2 layer 22222222 · p. 141
    • Other names(414121)P43No. 78C44P 4cwg96g97
    Extra links

  4. 142422

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g98: asymmetric motifs carry phase colours for Conway type 442/442.
    Space–time lift diagram for g98 and polar space group No. 79 I4; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine WOBr4.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · WOBr4No. 79 I4 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

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

    Relations Γ = ⟨α, β, γ | α⁴ = β⁴ = γ² = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(424021)I4No. 79C45I 4g98
    Extra links

  5. 444412

    Derived C4 short form — no literal book row

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 4-colouring for group g99: asymmetric motifs carry phase colours for Conway type 442⁴/2222.
    Space–time lift diagram for g99 and polar space group No. 80 I4_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine NbO2.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · NbO2No. 80 I41 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C4+1/4 period
    βquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C43+3/4 period
    γhalf-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone

    Relations Γ = ⟨α, β, γ | α⁴ = β⁴ = 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 (ACBD)
    • β (ADBC)

    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 242412 · p. 140C2 layer 12122222 · p. 141
    • Other names(434120)I41No. 80C46I 4bwg99
    Extra links

  6. 442

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g94: asymmetric motifs carry phase colours for Conway type 442.
    Space–time lift diagram for g94 and polar space group No. 75 P4; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Tetragonal Sodium Tungsten Bronze.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Tetragonal Sodium Tungsten BronzeNo. 75 P4 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    βquarter-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 | α⁴ = β⁴ = γ² = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(404020)P4No. 75C41P 4g94
    Extra links

∗442 6 forward groups

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

  1. ∗142412

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g129: asymmetric motifs carry phase colours for Conway type *442/*2222.
    Space–time lift diagram for g129 and polar space group No. 105 P4_2mc; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine BaGe2P2.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · BaGe2P2No. 105 P42mc CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionnonenone

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁴ = (QR)⁴ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗·4:4·2)P42mcNo. 105C4v7P 4c -2g129
    Extra links

  2. ∗241422

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g130: asymmetric motifs carry phase colours for Conway type *442/2*22.
    Space–time lift diagram for g130 and polar space group No. 101 P4_2cm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine [Me4N][Cu2(NCS)3].
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · [Me4N][Cu2(NCS)3]No. 101 P42cm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionC2+1/2 period
    Qmirror reflectionnonenone
    Rmirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁴ = (QR)⁴ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗:4·4:2)P42cmNo. 101C4v3P 4c -2cg130
    Extra links

  3. ∗242422

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g131: asymmetric motifs carry phase colours for Conway type *442/442.
    Space–time lift diagram for g131 and polar space group No. 103 P4cc; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine VOSe2O5.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · VOSe2O5No. 103 P4cc CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionC2+1/2 period
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁴ = (QR)⁴ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗:4:4:2)P4ccNo. 103C4v5P 4 -2cg131
    Extra links

  4. ∗141422

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g136: asymmetric motifs carry phase colours for Conway type *442/*442.
    Space–time lift diagram for g136 and polar space group No. 107 I4mm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Sm2Cu4Sn5.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Sm2Cu4Sn5No. 107 I4mm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone
    Rmirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁴ = (QR)⁴ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗·4·4:2)I4mmNo. 107C4v9I 4 -2g136
    Extra links

  5. ∗142422

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g138: asymmetric motifs carry phase colours for Conway type *442/4*2.
    Space–time lift diagram for g138 and polar space group No. 108 I4cm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Rb5Nb3OF18.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Rb5Nb3OF18No. 108 I4cm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁴ = (QR)⁴ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗·4:4:2)I4cmNo. 108C4v10I 4 -2cg138
    Extra links

  6. ∗442

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g128: asymmetric motifs carry phase colours for Conway type *442.
    Space–time lift diagram for g128 and polar space group No. 99 P4mm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine BaTiO3.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · BaTiO3No. 99 P4mm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone
    Rmirror reflectionnonenone

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁴ = (QR)⁴ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(∗·4·4·2)P4mmNo. 99C4v1P 4 -2g128
    Extra links

4∗2 6 forward groups

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

  1. 24∗22

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g133: asymmetric motifs carry phase colours for Conway type 4*2/22×.
    Space–time lift diagram for g133 and polar space group No. 106 P4_2bc; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Na(Zn(OH)3).
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Na(Zn(OH)3)No. 106 P42bc CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, P | α⁴ = P² = (PαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(42∗:2)P42bcNo. 106C4v8P 4c -2abg133
    Extra links

  2. 24∗12

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g134: asymmetric motifs carry phase colours for Conway type 4*2/2*22.
    Space–time lift diagram for g134 and polar space group No. 102 P4_2nm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine S4N2.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · S4N2No. 102 P42nm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C2+1/2 period
    Pmirror reflectionnonenone

    Relations Γ = ⟨α, P | α⁴ = P² = (PαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(42∗·2)P42nmNo. 102C4v4P 4n -2ng134
    Extra links

  3. 14∗22

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g135: asymmetric motifs carry phase colours for Conway type 4*2/442.
    Space–time lift diagram for g135 and polar space group No. 104 P4nc; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine HgTlBa2CuOx.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · HgTlBa2CuOxNo. 104 P4nc CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Pmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, P | α⁴ = P² = (PαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(40∗:2)P4ncNo. 104C4v6P 4 -2ng135
    Extra links

  4. 44∗12

    Derived C4 short form — no literal book row

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 4-colouring for group g137: asymmetric motifs carry phase colours for Conway type 4*2⁴/*2222.
    Space–time lift diagram for g137 and polar space group No. 109 I4_1md; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine NbP.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · NbPNo. 109 I41md CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Source-label note. The public model title says No. 108, but I4_1md is International Tables No. 109 and the CrystalSymmetry source page also identifies No. 109.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C4+1/4 period
    Pmirror reflectionnonenone

    Relations Γ = ⟨α, P | α⁴ = P² = (PαPα⁻¹)² = 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 (ACBD)

    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 24∗12 · p. 140C2 layer 22∗1212 · p. 140
    • Other names(41∗·2)I41mdNo. 109C4v11I 4bw -2g137
    Extra links

  5. 44∗22

    Derived C4 short form — no literal book row

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 4-colouring for group g139: asymmetric motifs carry phase colours for Conway type 4*2⁴/22×.
    Space–time lift diagram for g139 and polar space group No. 110 I4_1cd; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Be(BH4)2.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Be(BH4)2No. 110 I41cd CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C4+1/4 period
    Pmirror reflectionC42+1/2 period

    Relations Γ = ⟨α, P | α⁴ = P² = (PαPα⁻¹)² = 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 (ACBD)
    • P (AB)(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 24∗12 · p. 140C2 layer 22∗2222 · p. 140
    • Other names(41∗:2)I41cdNo. 110C4v12I 4bw -2cg139
    Extra links

  6. 4∗2

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g132: asymmetric motifs carry phase colours for Conway type 4*2.
    Space–time lift diagram for g132 and polar space group No. 100 P4bm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Na0.5 Bi0.5 TiO3.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Na0.5 Bi0.5 TiO3No. 100 P4bm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αquarter-turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Pmirror reflectionnonenone

    Relations Γ = ⟨α, P | α⁴ = P² = (PαPα⁻¹)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(40∗·2)P4bmNo. 100C4v2P 4 -2abg132
    Extra links

333 4 forward groups

A sphere with three order-3 cone points.

  1. 333333

    323232
    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 3-colouring for group g225: asymmetric motifs carry phase colours for Conway type 333³/◦.
    Space–time lift diagram for g225 and polar space group No. 145 P3_2; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Sheldrickite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · SheldrickiteNo. 145 P32 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αone-third 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.C32+2/3 period
    γone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C32+2/3 period

    Relations Γ = ⟨α, β, γ | α³ = β³ = γ³ = 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 (ABC)
    • γ (ABC)

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

    Identifications

    • The Symmetries of Things · p. 156
    • Other names(313131)P32No. 145C33P 32g226g225
    Extra links

  2. 333333

    313131
    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 3-colouring for group g226: asymmetric motifs carry phase colours for Conway type 333³/◦.
    Space–time lift diagram for g226 and polar space group No. 144 P3_1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Stillwellite-Ce.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Stillwellite-CeNo. 144 P31 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αone-third 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.C3+1/3 period
    γone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C3+1/3 period

    Relations Γ = ⟨α, β, γ | α³ = β³ = γ³ = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 156
    • Other names(313131)P31No. 144C32P 31g225g226
    Extra links

  3. 333313

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 3-colouring for group g227: asymmetric motifs carry phase colours for Conway type 333³/333.
    Space–time lift diagram for g227 and polar space group No. 146 R3; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Mg(SO3)(H2O)6.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Mg(SO3)(H2O)6No. 146 R3 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

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

    Relations Γ = ⟨α, β, γ | α³ = β³ = γ³ = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 156
    • Other names(303132)R3No. 146C34R 3g227
    Extra links

  4. 333

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g224: asymmetric motifs carry phase colours for Conway type 333.
    Space–time lift diagram for g224 and polar space group No. 143 P3; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Simpsonite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · SimpsoniteNo. 143 P3 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

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

    Relations Γ = ⟨α, β, γ | α³ = β³ = γ³ = αβγ = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(303030)P3No. 143C31P 3g224
    Extra links

∗333 2 forward groups

A mirror triangle with three order-3 corners.

  1. ∗232323

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g231: asymmetric motifs carry phase colours for Conway type *333/333.
    Space–time lift diagram for g231 and polar space group No. 158 P3c1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine RuCl3.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · RuCl3No. 158 P3c1 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionC2+1/2 period
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)³ = (QR)³ = (RP)³ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗:3:3:3)P3c1No. 158C3v3P 3 -2"cg231
    Extra links

  2. ∗333

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g230: asymmetric motifs carry phase colours for Conway type *333.
    Space–time lift diagram for g230 and polar space group No. 156 P3m1; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Fencooperite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · FencooperiteNo. 156 P3m1 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone
    Rmirror reflectionnonenone

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)³ = (QR)³ = (RP)³ = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(∗·3·3·3)P3m1No. 156C3v1P 3 -2"g230
    Extra links

3∗3 4 forward groups

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

  1. 13∗23

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g233: asymmetric motifs carry phase colours for Conway type 3*3/333.
    Space–time lift diagram for g233 and polar space group No. 159 P31c; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine LiNaSO4.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · LiNaSO4No. 159 P31c CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.nonenone
    Pmirror reflectionC2+1/2 period

    Relations Γ = ⟨α, P | α³ = P² = (PαPα⁻¹)³ = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(30∗:3)P31cNo. 159C3v4P 3 -2cg233
    Extra links

  2. 33∗13

    signature and kernel split across book rows

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 3-colouring for group g234: asymmetric motifs carry phase colours for Conway type 3*3³/*333.
    Space–time lift diagram for g234 and polar space group No. 160 R3m; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Tourmaline.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · TourmalineNo. 160 R3m CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C32+2/3 period
    Pmirror reflectionnonenone

    Relations Γ = ⟨α, P | α³ = P² = (PαPα⁻¹)³ = 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)

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

    Identifications

    • The Symmetries of Things · p. 156
    • Other names(31∗·3)R3mNo. 160C3v5R 3 -2"g234
    Extra links

  3. 33∗23

    Derived C6 short form — no literal book row

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 6-colouring for group g235: asymmetric motifs carry phase colours for Conway type 3*3⁶/333.
    Space–time lift diagram for g235 and polar space group No. 161 R3c; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Proustite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · ProustiteNo. 161 R3c CrystalSymmetry exampleOpen model on SketchfabBilliards animation Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    αone-third turnCounterclockwise in the canonical face-order convention; some plates use a reflected setting.C64+2/3 period
    Pmirror reflectionC63+1/2 period

    Relations Γ = ⟨α, P | α³ = P² = (PαPα⁻¹)³ = 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 (ABD)(CEF)
    • P (AC)(BE)(DF)

    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 13∗23 · p. 140C3 layer 333313 · p. 156
    • Other names(31∗:3)R3cNo. 161C3v6R 3 -2"cg235
    Extra links

  4. 3∗3

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g232: asymmetric motifs carry phase colours for Conway type 3*3.
    Space–time lift diagram for g232 and polar space group No. 157 P31m; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine Galeite.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · GaleiteNo. 157 P31m CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

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

    Relations Γ = ⟨α, P | α³ = P² = (PαPα⁻¹)³ = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(30∗·3)P31mNo. 157C3v2P 3 -2g232
    Extra links

632 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
    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    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.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · Sr(S2O6)(H2O)4No. 171 P62 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    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
    Extra links

  2. 363312

    p. 156 typo corrected by pp. 157 and 164

    64312
    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    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 Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    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
    Extra links

  3. 261322

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    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.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · NephelineNo. 173 P63 CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    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
    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    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 Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via 3Dmol.js + COD.

    Presentation

    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
    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    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 Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    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

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    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 Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    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

∗632 4 forward groups

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

  1. ∗162322

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g269: asymmetric motifs carry phase colours for Conway type *632/3*3.
    Space–time lift diagram for g269 and polar space group No. 185 P6_3cm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine KNiCl3.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · KNiCl3No. 185 P63cm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁶ = (QR)³ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗·6:3:2)P63cmNo. 185C6v3P 6c -2g269
    Extra links

  2. ∗262322

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g270: asymmetric motifs carry phase colours for Conway type *632/632.
    Space–time lift diagram for g270 and polar space group No. 184 P6cc; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine AlPO-5.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · AlPO-5No. 184 P6cc CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionC2+1/2 period
    Qmirror reflectionC2+1/2 period
    Rmirror reflectionC2+1/2 period

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁶ = (QR)³ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗:6:3:2)P6ccNo. 184C6v2P 6 -2cg270
    Extra links

  3. ∗261312

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 2-colouring for group g271: asymmetric motifs carry phase colours for Conway type *632/*333.
    Space–time lift diagram for g271 and polar space group No. 186 P6_3mc; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine AgI.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · AgINo. 186 P63mc CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionC2+1/2 period
    Qmirror reflectionnonenone
    Rmirror reflectionnonenone

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁶ = (QR)³ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 140
    • Other names(∗:6·3·2)P63mcNo. 186C6v4P 6c -2cg271
    Extra links

  4. ∗632

    JavaScript is needed for this film; the static coloured plate remains available below.

    Loading local film data…

    0.000
    Static perfect 1-colouring for group g268: asymmetric motifs carry phase colours for Conway type *632.
    Space–time lift diagram for g268 and polar space group No. 183 P6mm; exact clock phases form stacked sheets in vertical time through one clock period. Activate the interactive viewer to examine AuCN.
    Opens the public Sketchfab WebGL viewer externally. Only one crystal viewer stays live at a time.
    3D crystal · AuCNNo. 183 P6mm CrystalSymmetry exampleOpen model on Sketchfab Before loading, this space–time lift shows the clockwork wallpaper; the vertical axis is time through one clock period. The interactive pane is the real-crystal exemplar via Sketchfab.

    Presentation

    GeneratorColorTime
    Pmirror reflectionnonenone
    Qmirror reflectionnonenone
    Rmirror reflectionnonenone

    Relations Γ = ⟨P, Q, R | P² = Q² = R² = (PQ)⁶ = (QR)³ = (RP)² = 1⟩

    Identifications

    • The Symmetries of Things · p. 40
    • Other names(∗·6·3·2)P6mmNo. 183C6v1P 6 -2g268
    Extra links

Data

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