Catalogue C — the clockwork crystals

The 68 spacetime groups in which time never runs backwards: 17 trivial clocks and 51 groups that genuinely entangle space with time — each one a cyclic colouring of a wallpaper pattern, each one the shadow of a 3D space group.

Fix the ambient group E(2) × E(1) of Xu–Wu's Eq. (4) — operations Γ(r, t) = (Rr + u, st + τ) — and keep only the discrete subgroups all of whose elements have s = +1: no symmetry plays the film backwards. Counted up to time-preserving affine equivalence, which includes the Galilean boosts (r, t) ⟼ (r + tw, t), exactly 68 of the 275 spacetime groups are forward. Each is a wallpaper group wearing a clock that only turns one way. Seventeen wear it trivially — direct products wallpaper × clock, one per wallpaper group, in which no spatial symmetry touches the clock — and the remaining 51 genuinely entangle space with time: some rotation, glide or lattice translation cannot be performed without also advancing the clock.

Because the group is forward, the map "how far does this symmetry advance the clock" is a homomorphism onto a finite cyclic group ℤ/k, with k ∈ {1, 2, 3, 4, 6}. So a clockwork crystal is a cyclic colouring of its wallpaper projection: paint each copy of the motif by its phase, one colour per k-th of the period, and every symmetry shifts the k colours cyclically. The plates below are exactly this picture — colour = phase. Catalogue D works the correspondence out in full, including which cyclic colourings are not realised by any clock — the boostable ones, whose colour advance rides on a translation that a Galilean boost absorbs — and the four enantiomorphic clock pairs that drive the same colouring: a clock and its reversal stack the same colours in opposite order.

There is a second reading of the same data. Read the clock phase as a height — lift each motif copy to the altitude given by its phase, periodically continued — and a clockwork crystal becomes a crystal in ordinary 3D space. This is Belov and Tarkhova's 1956 construction run in reverse: they projected stacked layers down to coloured mosaics, colour as a code for height. Under this lift the 51 nontrivial clockwork groups land on 51 distinct space groups among the 230 of catalogue B — the order-2 clock on p4 becomes the 4₂ screw of P4₂, the order-6 clock on p6 the 6₁ screw of P6₁ — while the 17 trivial clocks land on the 17 vertical stacks wallpaper × ℤ, from P1 up to P6mm. The four enantiomorphic clock pairs lift to four of the eleven enantiomorphic space-group pairs: running the clock backwards mirrors the screw.

How to read an entry: the lead symbol is the colour-superscript orbifold signature of The Symmetries of Things (decoding rules in the notation guide) — a superscript on an orbifold feature is the order of the clock advance that feature induces. The line beneath gives the colour type G/K: the wallpaper group G of the projection and the kernel K, the subgroup of symmetries that leave the clock alone, both linked into catalogue A. The plate draws one comma motif per group element, coloured by phase; the remaining links lead to the live animation, the colouring in catalogue D, the space-group lift in catalogue B, the reverse-clock mate where the clock has a handedness, and the companion-site atlas entry.