Spacetime groups: the crystallography of looping animations

A catalog and tutorial for the symmetry groups of animations that loop in time and tile the plane — after T. J. Fletcher (1956) and Xu–Wu (2018).

Every animation on this site starts paused. Click one to play or pause it (space does the same once you have clicked), or point at a frame and jump between its symmetry instants with ← → — hold shift to step frame by frame instead. ⏮ returns the loop to t = 0.

Reading a diagram. The motif is a comma — an asymmetric, handed shape, so a rotated copy looks rotated and a reflected copy looks reflected. It fills like a vessel once per period: its fill level is its internal time. The ring around it is a phase visualiser: the group's time operations cut the period into a whole number of intervals, and the ring shows one arc per interval. A hand sweeps round it at constant speed, one turn per period, and its point marks the copy's phase — so which interval the copy is in is read off from where the point currently is. Rings are drawn upright on the screen and are never turned with their copy, so the same interval is the same arc everywhere and a staircase of phases can be read off at a glance. Which way the hand sweeps is that copy's direction of time — clockwise where it is filling, anticlockwise where a time reversal has it draining, which is the one thing a still frame cannot otherwise show. The same intervals are marked under the scrub bar — click a mark to jump to it (◆ marks an instant about which the loop is a palindrome).

1. Animations as spacetime crystals

An animation is a function on ℝ² × ℝ — the plane times time. If it loops with period T and tiles the plane with a lattice of translations, its symmetry group is a crystallographic group of 2+1-dimensional spacetime. Fletcher's 1956 observation, made for mathematical animations, is that one can treat t as a third spatial coordinate and reuse the theory of space groups. One constraint distinguishes spacetime from ℝ³: no symmetry may rotate a space direction into the time direction. Formally, a spacetime group is a discrete subgroup G of E(2) × E(1) — each element acts by

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

— such that the translations in G span a rank-3 lattice containing a pure time translation (the loop) and a rank-2 spatial lattice (the tiling). The element with s = −1 plays the animation backwards; τ is a fraction of the period. These are the spacetime groups of Xu and Wu (2018), the symmetry groups of a spacetime crystal; Fletcher, who got here first and was not read, called them film groups — see History.

Two spacetime groups are equivalent when conjugate by an affine map of spacetime that preserves absolute time — x ↦ Ax + ut + b, t ↦ ct + d — with 3D orientation preserved (det A · c > 0, the convention under which the crystallographic screw pairs 4₁/4₃ count as two groups). The allowed shear u is a Galilean boost: switching to a camera that pans with constant velocity. Boosts matter: a wallpaper pattern drifting steadily across the screen is, as a spacetime crystal, indistinguishable from the same wallpaper standing still — pan the camera along with it. Drift is never new symmetry. What cannot be boosted away is the subject of this site.

2. Warm-up: one spatial dimension

Take one space dimension and time — a looping animated strip. The possible point-like operations form at most a Klein four-group: the spatial mirror m_x, the time mirror m_t (the loop is a palindrome), and their product, the 2-fold spacetime rotation (flip space and reverse time). Combined with fractional translations these give exactly 13 groups (Xu–Wu 2018) — the spacetime-group analogue of the 7 frieze groups. All thirteen, live (each comma fills once per period — its fill level is its internal time — so phase differences are visible at a glance, and time-reversed copies drain while the others fill):

Three phenomena appear here that persist in 2+1D. The time glide (the tilded mirrors of ~*~*): mirror symmetry holds only after waiting half a period. The glide time-reversal (the clause of o/g′): the animation played backwards equals itself shifted half a cell. And time centring (the c-prefixed groups): the spacetime lattice is centred, so each motif's nearest neighbours run half a period out of phase. Fletcher's 1956 paper counted 7 groups at this stage rather than 13 — see History for the two (instructive) reasons.

3. The operations, in 2+1D

In two spatial dimensions the point operations are rotations of order 1, 2, 3, 4, 6 and reflections, each optionally composed with time reversal, and each dressed with a fractional spacetime translation. The genuinely new operations, in Xu–Wu's terminology:

Everything else is inherited: pure spatial glides, ordinary rotations and mirrors acting frame-by-frame, and the magnetic-group operations (reversal composed with a spatial isometry) familiar from Shubnikov theory with "colour swap" read as "play backwards".

4. The classification

The proof is finite linear algebra, and the computation is in this repository. Fix the magnetic point group P (spatial op, time sign) and a compatible spacetime lattice L; the possible systems of fractional translations form the cohomology group H¹(P, ℝ³/L), and the spacetime groups in this arithmetic class are its orbits under the normaliser (lattice re-basings, boosts, and combined orientation flips). Summing orbits over the 72 arithmetic classes gives 275. The same machinery run in 1+1D yields the 13 strip groups above, and run on static 2D patterns yields the 17 wallpaper groups — two anchors we use as validation, along with the per-system totals of Xu–Wu's Table II (which for the tetragonal, trigonal and hexagonal systems equal the 3D space-group counts 68, 25, 27, since there a spacetime group is a space group with distinguished c-axis).

Structurally: a spacetime group G determines a wallpaper group Γ (its spatial projection — the symmetry of the long-exposure photograph) and each spatial operation carries a clock phase; quotienting spacetime by G gives a 3-orbifold fibred in time-circles over Γ's orbifold. Conway and Goodman-Strauss's orbifold notation, with the superscripts that The Symmetries of Things uses for colourings, records exactly this Seifert-fibration data and names every group in the catalog; products (wallpaper × clock) are precisely the entries with no decorations.

5. Explore

Catalog — all groups, filterable, each with a live animation, generators, orbifold symbol and Hermann–Mauguin-style name.
Gallery — the featured non-product groups with downloadable looping GIFs: time screws, time glides, time centrings, and palindromes, each tied to the paper's terminology.
Notation — the decoration rules, with the Seifert/fibrifold justification.
The hexagon — cosets, stabilisers and kernels worked out on D₆, with every colouring of a hexagon drawn; the finite rehearsal for the coloured crystals.
Subgroups versus colour groups — are the subgroups of a wallpaper (or hyperbolic) group the same thing as its colourings? The dictionary, the counts under each equivalence, and the simplest counterexamples.
History — Fletcher 1956 (a forgotten precursor with zero citations), Shubnikov's antisymmetry, Janssen–Janner–Ascher, choreographic crystals, and the H/K theorem of equivariant dynamics.

6. Crystals

Step back from spacetime for a moment and ask what the word crystal was doing all along. Every classification met on this site is an instance of one two-part recipe. Fix an ambient group 𝒜 — all the transformations a symmetry is allowed to be — and an equivalence relation: a prescribed family of conjugations that says when two symmetry groups count as the same. A crystal is then a discrete subgroup G ≤ 𝒜 whose translations form a full-rank lattice, and a crystal type is an equivalence class of such subgroups. Xu–Wu's Eq. (4) is the spacetime instance: operations

Γ(r, t) = (Rr + u, st + τ)

— a spatial isometry paired with an action on an internal coordinate, the clock — with equivalence given by time-preserving affine maps, boosts included (§1). Vary the two ingredients and the neighbouring crystallographies appear, each with its own finite census. Swap the internal coordinate for nothing: the wallpaper groups. For a third spatial dimension: the space groups. For a finite set of k interchangeable labels acted on by permutations — replace the clock by a colour — and the colour groups appear, with the clock case embedded as the colourings a clock can drive:

The catalogues are cross-linked entry by entry, and the boundaries carry the interesting theorems. Every one of the 68 clockwork crystals is a cyclically coloured wallpaper (colour = phase), but not every cyclic colouring is a clockwork crystal: a colour that advances along a translation is boostable away, which is why catalogue C is finite while colourings of p1 exist for every k. The 51 nontrivial clockwork groups lift, phase as height, to 51 of the 230 space groups of catalogue B — Belov's 1956 stacking construction, run in reverse. And the one coloured crystal that is chiral (the five-colouring of p4) is the 2D shadow of the enantiomorphism that separates 230 space groups from 219.

A · 2D crystals — the 17 wallpaper groups as crystals: subgroups of E(2) up to proper affine equivalence, with each group's colourings, clockwork lifts and vertical stack.
B · 3D crystals — the 230 space groups, one worked mineral structure each, with the enantiomorphic pairs and the clockwork lifts marked.
C · clockwork crystals — the 68 forward spacetime groups as coloured wallpapers: parent, colour-preserving kernel, space-group lift, and the colouring each one drives.
D · coloured 2D crystals — all 269 colourings of the wallpaper groups in at most six colours, recomputed from scratch and rendered; with the census table, the clockwork correspondence, and the chirality story.
E · colour pattern types — the finer Grünbaum–Shephard classification: 88 + 59 periodic types for two and three colours, the Figure 8.3.2 wedge that forces the refinement, and the open problem at k = 4.
F · Two-Color Patterns — the 88 two-colour pattern types recomputed from scratch, one plate each with Chaim's presentation and colour permutations, and the invariant that separates types over one colour group: the marked signature (where the motif sits), with crops of both books.

References

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 — the source of this catalog: the 13 groups in 1+1D and the 275 in 2+1D (appendix C, Table II).
C. Ke, C. Wu, Two-Dimensional Space-Time Groups: Classification and Applications (2026). arXiv:2604.05619 — a later re-derivation of the same 275, with the chirality-selective response rule and the horizontal cone cited in the gallery.
J. H. Conway, O. Delgado Friedrichs, D. H. Huson, W. P. Thurston, On Three-dimensional Space Groups, Beitr. Algebra Geom. 42 (2001) 475–507. arXiv:math/9911185
J. H. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries of Things, A K Peters, 2008.
P. Scott, The geometries of 3-manifolds, Bull. LMS 15 (1983) 401–487.