The hierarchy

Coloured wallpaper groups, spacetime groups, clockwork groups, space groups, polar space groups: the maps between them, with counts, for at most six colours.

A doubly periodic animation that loops in time is classified by several established schemes with different counts. This page fixes the maps between them and records which are injective, surjective or bijective. Two are exact.

Each arrow is labelled with what it forgets; only the bottom one is a bijection. Two things the diagram settles. Spacetime groups are not a subset of the coloured wallpaper groups — only the clockwork ones are colourings, since a colour permutation cannot reverse the palette (§5). And they are not merely comparable with the space groups: every spacetime group is a space group, necessarily non-cubic, and what fails is injectivity (§6).

1. The classifications

All describe a pattern periodic in two spatial directions, and differ in the extra datum carried and in the equivalence used.

Two definitions. A coloured wallpaper group (Wieting's chromatic plane ornament, Senechal's colour group) is a wallpaper group Γ with a transitive action on N colours, equivalently a subgroup of index N; two agree when an affine map of the plane carries one to the other with colours permuted freely. A polar space group is one whose point group fixes a direction together with its sense — the ten pyroelectric classes 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm.

2. Phase, colour, height

An element of a spacetime group acts by

(x, y, t) ⟼ (M(x, y) + v,  st + τ),   M ∈ O(2),  s = ±1,

and τ admits three readings. As a phase it is a fraction of the period. As a colour it is one of N labels, available when s ≡ +1 and the τ occurring form a cyclic group of order N. As a height it is a fractional translation along a third axis, making the animation a crystal in ℝ³. The sign s is what distinguishes time from a third spatial coordinate. Under the height reading:

Three caveats, developed below: the height reading forgets which axis was time (§6); the colour reading requires s ≡ +1 (§5); and the three subjects use different equivalences (§8).

3. Colour, up to six

Fix a wallpaper group Γ. A homomorphism χ: Γ → ℝ/ℤ with image of order N assigns each element a phase; ker χ is normal of index N with cyclic quotient, and its cosets are the N colours. A clockwork group is exactly such a pair (Γ, χ) — a coloured wallpaper group whose colour group is C_N acting regularly.

Three notions of colouring, N = 1…6:

Column 1 is not monotone in N. Column 2 restricts to regular cyclic colour groups, column 3 to those realised by distinct spacetime groups.

The same argument bounds N: a clock has order 1, 2, 3, 4 or 6. Six colours is the full range, not an imposed cut-off. The five clocks:

4. The image of the clock

Each element acts on S¹ = ℝ/ℤ as t ↦ t + τ or t ↦ −t + τ, so the image of G in Isom(S¹) is finite: C_N if G has no time reversal, D_N otherwise. All 275, by that image:

The cyclic row is column 3 of §3. The 17 with C₁ are the wallpaper groups with an independent loop.

5. Which spacetime groups colour their projection

Write Φ: G → Isom(S¹) for the action on the loop, and Γ for the spatial projection. A colouring of Γ is what Φ induces when it descends along G ↠ Γ, and it descends exactly when ker π ⊆ ker Φ — that is, when no element of G acts trivially on space and non-trivially on time. The colours are then the cosets of ker Φ and the colour group is the image. Four cases:

The second row answers the obvious question about the first. D₁ is cyclic of order 2, so those … groups have a regular cyclic colour group and yet reverse time: the colour is the direction of time, not a phase, and the colouring is a black-and-white one in Shubnikov's sense. In total … of the 275 induce a regular cyclic colouring, of which only the 68 clockwork groups colour by phase. A clock is a regular cyclic colouring whose colour group acts on the loop by rotations; an antisymmetry is one whose generator acts by a reflection.

The last row is the case with no induced colouring at all: G contains a pure time reversal, an element (x, t) ↦ (x, −t + τ₀) that plays the animation backwards without moving anything. Φ is then non-trivial on ker π and does not descend.

6. Forgetting which axis is time

Reading each phase as a height makes a spacetime group a discrete group of isometries of ℝ³ with a rank-3 lattice, that is, a space group.

No cubic type occurs because a cubic point group fixes no direction, whereas the time axis is fixed by every spacetime-group operation. Every non-cubic type occurs because a non-cubic point group fixes some axis, which may be taken as time. Fletcher subtracted the 36 cubic groups from 230 and obtained 194 spacetime groups; the image is right and the count is not, since the map is not injective — one crystal may be sliced into inequivalent animations according to which invariant axis is time and whether the operations reversing it are read as mirrors or as time reversals.

Tetragonal, trigonal and hexagonal have a unique high-order axis, so spacetime groups and types agree: 68, 25, 27. Monoclinic splits in two, according to whether the 2-fold axis is temporal or spatial (the catalog's T- and R-monoclinic). Orthorhombic has three perpendicular axes and splits by up to three. The fibres sum to 275; the types number 194.

One fibre: …, three inequivalent animations.

7. Clockwork groups are the polar space groups

Restrict the map to groups without time reversal. In the height reading this says that no operation sends z ↦ −z: the point group fixes the z-axis together with its sense, which is the definition of a polar space group.

The point groups in the table below are therefore the ten polar classes, which are the ten planar point groups C₁, C₂, C₃, C₄, C₆, D₁, D₂, D₃, D₄, D₆ under their 3D names. Physically: a polar crystal admits a spontaneous polarisation along its polar axis, and a clockwork animation has a preferred direction of time, so its reverse has a different symmetry group. The two conditions coincide.

All 68:

The screw axes and glide planes are the time operations of §2: P4₁ is the quarter-period time screw, P6₁ the sixth-period one, the c-glides of P4cc and P6cc are time glides, and the rhombohedral centrings of R3 and R3c are time centrings. Eight rows form four enantiomorphic pairs (…): equivalent as coloured patterns, since a reflection of the plane exchanges them, but inequivalent as animations, since no change of frame sends a clock advancing by +1/N to one advancing by −1/N.

8. Where the counts differ

Comparing columns 2 and 3 of §3 group by group: the totals shrink over four wallpaper groups and grow over three. Both directions are accounted for.

Losses: the phase a change of frame removes

A phase carried by the translations is a homomorphism χ|_L: L → ℝ/ℤ, and it is P-invariant, so it factors through the coinvariants L_P = ℤ²/⟨(1 − g)ℤ² : g ∈ P⟩. A change of frame subtracts from χ the character ℓ ↦ w·ℓ — a Galilean boost x ↦ x + wt, or a re-slicing of simultaneity t ↦ t + w·x — and such a shear keeps the group inside E(2) × E(1) only when w is fixed by P.

Over the thirteen groups with L_P finite nothing is removable, and there the colouring count is never larger than the spacetime-group count. Over p1, pm, pg and cm the free part absorbs phase, and … colourings realise … spacetime groups:

Examples of colourings that are not spacetime groups.

p1, every N ≥ 2. L_P = ℤ², so every character is removable. Colour the cell in column k by k mod N: a striped N-colouring of p1, and a perfectly good one. As an animation it is the wallpaper sliding one cell every N-th of a period, and a camera panning at that speed sees it standing still. Five colourings, no new group.

pm, pg and cm at N = 3. Their coinvariants are ℤ × ℤ/2, ℤ × ℤ/2 and ℤ, the free factor being the translation along the mirrors. An order-3 character has nowhere else to live — the point group has order 2 and the torsion has exponent 2 — so the unique 3-colouring of each is striped along the mirror direction, and drifts. Three more colourings, no new group.

Every five-colouring there is. Five-colourings exist only over those same four groups, one each. Elsewhere an order-5 character would have to restrict to L_P as either trivial or onto C₅; the first makes it a character of P, and no point group has order divisible by 5, while the second needs an element of order 5 in L_P, whose exponent is 2, 3 or 4 in the table above. All four that exist are drifts. That is the whole of the empty N = 5 row of §3, and N = 5 is the only order whose row is empty.

pm at N = 4. Three colourings. Order 4 again needs the free ℤ; subtract that drift and what remains has order at most 2 — a colouring already counted higher up the same column. This is the general shape of the remaining losses: not that the colouring fails to be an animation, but that the animation it gives is one already listed with fewer colours.

Relabelling

A second, smaller effect: the height reading can exchange a time offset for a spatial one. Giving pm's mirror half a period lifts it to a glide plane, type Pc (7), which is also the lift of pg's ordinary spatial glide; under the crystallographic convention used here, in which any unimodular re-basing of the spacetime lattice is allowed, these are one spacetime group. So the 42 colourings of the four rotation-free groups realise five spacetime groups in all — P1, Pm, Pc, Cm and Cc — but not with the base assignment the colour census would predict.

Splits: chirality

The smallest case is 333 = p3: three cyclic colourings, four clockwork groups. All four, each with the static colouring it induces on the plane —

The animation is this catalog; the static plate beside it is the same group's perfect colouring, from the companion clockwork/colouring correspondence, where all 68 are plotted.

3₁3₁3₁ and 3₂3₂3₂ are one colouring and two spacetime groups. Compare their plates: the same motifs in the same places, with two of the three colours exchanged. That is not a coincidence but the definition. A cyclic colouring is a normal subgroup K ⊴ Γ of index N with cyclic quotient, the colours being its cosets, and χ and χ⁻¹ have the same kernel. A colouring therefore records the partition into colour classes and not which way round the cycle runs — which is why the correspondence tables give the two the same signature ³3³3³3 and the same colour type 333³/◦.

A spacetime group does record it, because the phase circle is oriented: around a 3-centre the copies come at 0, ⅓, ⅔ in one and 0, ⅔, ⅓ in the other, and that is the difference between a film and the same film run backwards. The change of frame that would undo it is a reflection of the plane, which sends χ(R) = +1/3 to −1/3 and P3₁ to P3₂. As a map of coloured patterns that is an equivalence, and the colours come along as labels. As a change of frame it is inadmissible on its own: det A · c > 0 forces it to reverse time as well, and the composite sends χ(R) = +1/3 back to +1/3 — to P3₁ again. No allowed frame identifies the two, so the pair survives.

The third colouring, 333³/333, is the one whose phase sits on the translations: a tripled cell, realised by the single group r33₁3₂ = R3. It carries a 3₁ and a 3₂ axis at once, so its own mirror image is itself and it does not split. Four colouring classes across p3, p4 and p6, eight spacetime groups: exactly the four enantiomorphic pairs among the 68 polar types.

The same argument, verbatim, over p4 (P4₁/P4₃) and p6 (P6₁/P6₅ and P6₂/P6₄). Two conditions are needed, and between them they pick out those three bases. The point group must contain no reflection, or the reflection is already inside the group and conjugates χ to χ⁻¹ for free — which rules out the twelve bases with a mirror. And the clock must have order N ≥ 3, since for N ≤ 2 the two senses ±1/N coincide — which rules out p1 (everything drifts) and p2 (no clock beyond order 2). Inside p3, p4 and p6 the pairs are the groups that do not already contain both senses: R3 and I4₁ carry a 3₁ and a 3₂, a 4₁ and a 4₃, so they are their own mirror images and stay single. That is why p3 and p4 gain one apiece and p6 gains two.

9. Adjacent classifications

Magnetic (Shubnikov) groups. The 80 magnetic plane groups (17 + 17 grey + 46 black-white) and the 1651 magnetic space groups (230 + 230 grey + 1191 black-white) are the two-colour theory, with the colour swap originally read as time reversal: the N = 2 case here. A D₁ clock is antisymmetry; a C_N clock is Zamorzaev's P-symmetry with quality group ℤ/N. Animation groups add that the colour set is the circle ℝ/ℤ, in which ℤ/N sits as the image of χ, and that the circle is oriented.

Subperiodic groups. The 7 frieze groups (plane, rank-1 lattice) and the 75 rod and 80 layer groups (space, rank-1 and rank-2) classify lattices that do not fill their space. They cover animations that do not loop, or tile in one direction only; spacetime groups have a rank-3 spacetime lattice and are space groups. The 1+1D analogue of the friezes is the 13 strip groups, counted on the tutorial.

Sohncke and chiral space groups. The 65 Sohncke groups contain no orientation-reversing operation, and 22 of them, in 11 enantiomorphic pairs, are chiral. This cuts the 230 differently from polarity: polarity fixes a direction, chirality an orientation, and neither class contains the other. In spacetime terms Sohncke means no spatial mirror at any phase, polar means no time reversal.

Choreographic crystals. Boyle–Khoo–Smith classify choreographic order — finite configurations whose symmetries are spatial operations composed with time offsets — which is the clockwork construction without a spatial lattice. The Golubitsky–Stewart H/K theorem describes which (setwise, pointwise) pairs a time-periodic state realises. Both concern the C_N quotient of §5.

References

T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments, Dekker, 1982 (Table 11 = OEIS A307293).
M. Senechal, Color groups, Discrete Appl. Math. 1 (1979) 51–73.
J. D. Jarratt, R. L. E. Schwarzenberger, Coloured plane groups, Acta Cryst. A36 (1980) 884–888. doi:10.1107/S0567739480001866
A. M. Zamorzaev, Generalisation of the Fedorov groups, Kristallografiya 2 (1957) 15–20 — the 1651 antisymmetry space groups, first derived in his 1953 thesis.
T. J. Fletcher, Film Groups, Math. Gazette 40 (1956) 15–19. doi:10.2307/3610262
S. Xu, C. Wu, Space-Time Crystal and Space-Time Group, PRL 120, 096401 (2018). arXiv:1703.03388
C. Ke, C. Wu, Two-Dimensional Space-Time Groups: Classification and Applications (2026). arXiv:2604.05619
J. H. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries of Things, A K Peters, 2008 (colour signatures).
International Tables for Crystallography, Vol. A (space-group numbering); space-group types identified with spglib.
L. Boyle, J. Y. Khoo, K. Smith, Symmetric satellite swarms and choreographic crystals, PRL 116, 015503 (2016). arXiv:1407.5876

All counts here are regenerated from the catalog by enumerate/hierarchy.py, which asserts the theorems of §6 and §7 and re-derives the space-group types under --verify. The companion clockwork/colouring correspondence and cyclic-colouring/polar-space-group atlas give the same 68 records with static coloured plates and cell presentations.