Future directions: coloured groups + clockwork groups

A spacetime group is a wallpaper group coloured by a circle of phases. The discrete theory of colour symmetry is the natural companion — and the natural next step — of the clockwork orbifold notation.

The forward-time spacetime groups have a name here: a clockwork group is a spacetime group whose every symmetry runs time forwards, equivalently a wallpaper group in which every operation advances the animation by a fixed fraction of the period — a pattern with a clock, and a clock that only turns one way. They are the 68 groups whose clockwork orbifold symbol carries no prime clause, and the ones a colouring can express.

1. Colour symmetry in one paragraph

A perfect colouring of a wallpaper pattern assigns each copy of the motif one of N colours so that every symmetry of the uncoloured pattern permutes the colours consistently: following van der Waerden and Burckhardt, a colouring is a homomorphism φ from the wallpaper group Γ onto a transitive group of colour permutations, two colourings being equivalent when the pairs (Γ, φ) are. For two colours the data is just the index-2 subgroup H = ker φ of colour-preserving operations, and the classical count is 46 two-coloured (black–white, dichromatic) plane types, alongside the 17 uncoloured and the 17 grey types in which the swap commutes with everything — 80 Shubnikov classes in all. The 46 split further into 28 whose swaps are point operations and 18 whose swap is an anti-translation (Belov's p_b′-type coloured lattices) — the exact two-colour ancestors of the c and r stacking prefixes. The subject is older than it looks: Alexander and Herrmann's eighty two-dimensional layer groups (1929) match the 80 dichromatic classes one-for-one, reflection through the layer plane playing colour swap — precisely the trick animation groups play with time.

Two notations coexist. Crystallographers, after Belov and Tarkhova, prime the colour-reversing operations (p4′g′m); The Symmetries of Things (Part II, chapters 11–13) instead names a colouring by the orbifold pair full group / colour-preserving kernel — Escher's threefold reptiles are its 632//333. The clockwork notation's reversal clause is the ToS pair read backwards: o/m′[*×] lists the kernel with its clocks, then the full projection in the bracket — the pair *× / o.

2. Two colours, first way: play it backwards

Time reversal is a two-colouring of spacetime: the forward part G⁺ sits inside a spacetime group G with index 2, exactly as the colour-preserving subgroup sits inside a dichromatic group. Paint every copy of the motif by its time sign and the animation becomes a black–white pattern; "swap the colours" reads "play the animation backwards". The example below is o/g′, glide time-reversal: the swap is carried by a half-cell translation, so black and white columns alternate — one of the classical translation-swap ("p_b′") colourings.

Restricted to the spacetime groups with no clock at all — every forward operation instantaneous, no time-centred lattice — this correspondence is exact enough to count. The catalog contains … grey classes (the products /1′) and … proper clockless reversal classes. Against the classical 46 the proper count reads 45, and the missing pair is a theorem, not an accident. The one absent type is (cmm, cm) — Belov's cmm′ — and it is not missing from the catalog, only from the clockless census: re-base the spacetime lattice, trading the colour-preserving translation axis for time, and its centred spatial lattice becomes a time-centring. The catalog files it as the clocked class c**/m′[*2222]. This is the reversal-layer instance of the mechanism that files ~*~* under ××; the strict frame-preserving classification (283 classes) separates the two readings again.

3. Two colours, second way: wait half a period

Colour symmetry does not care whether the swap is realised by an orientation-reversing operation of time or by time itself. A spacetime group whose clock takes only the values 0 and ½ paints the same two colours unitarily: copies at phase 0 in one colour, copies at phase ½ in the other, and the swap is "wait half a period". Time-centred lattices are the purest case — c222₁2₁ is the checkerboard whose two sublattices trade colours every half period — and the 2₁ screws and tilded mirrors are the point-operation swaps. The data is once more a pair (Γ, index-2 kernel), so this is a second animation reading of the dichromatic types; the catalog has … forward classes with clock denominator 2 rather than 46, because the animation equivalence coarsens the translation-supported types — the same convention gap §4 quantifies.

4. N colours: the clock as a paint pot

Now let the colour group be cyclic of order N. A perfect ℤ_N-colouring of Γ is a homomorphism φ: Γ → ℤ_N; a forward spacetime group whose offsets have denominator N is literally this datum, with φ(g) = N·τ(g) mod N. Sample the animation at N equally spaced instants and paint each copy by the instant at which it is, say, full: a time screw n_k becomes a gyration that advances the colour by k — Escher's trick of rotating a pattern into its own recolouring. The static colourings below are computed from the animation specs themselves (hue = phase): four colours cycling around every 4-centre of 4₁4₁2₁, six around every 6-centre of 6₁3₁2₁.

5. Counting, side by side

The forward layer of the catalog therefore is a census of cyclic colour symmetries of the plane, and can be set against the classical one. Wieting enumerated the colour plane groups of every index N ≤ 60 by Senechal's congruence method; his Table 11 totals are preserved as OEIS A307293, and begin 17, 46, 23, 96, 14, 90 — the first two entries being the 17 wallpaper groups themselves and the 46 dichromatic types. Only N ∈ {1, 2, 3, 4, 6} can occur for an animation: a clock denominator is the order of a rotation, so the crystallographic restriction applies to colours as well.

The three rows differ for three separate and identifiable reasons, and the per-group table makes each visible. Wieting's column counts every transitive colour group of degree N; an animation sees only the cyclic ones. Animation equivalence adds Galilean boosts and spacetime re-basings, which absorb colourings carried by translations. And the animation classification keeps 3D orientation, so it is finer in one respect: a chiral pair of animations is a single colouring, since relabelling colours by k ↦ −k turns one into the other. Quotienting by that relabelling is the second row, and it merges exactly four pairs — 4₁4₁2₁/4₃4₃2₁, 3₁3₁3₁/3₂3₂3₂, 6₂3₂2/6₄3₁2, 6₁3₁2₁/6₅3₂2₁.

For N ≥ 3 the gap is dominated by the non-cyclic colour groups that a clock cannot express. Wieting's 23 three-colourings include, for example, the single three-colouring of 2222 (p2), whose colour group is necessarily S₃: a half-turn inverts every translation, so any homomorphism onto ℤ₃ kills the lattice, and p2's three-colour cell is correctly empty. The argument needs N odd — at N = 2 the half-turn's inversion is invisible, and p2 has two clockwork groups with a live clock, 2₁2₁2₁2₁ and c222₁2₁, which is what the row's N = 2 entry counts. Where the colourings are cyclic the two censuses meet again: 333 (p3) has two published three-colourings, and the catalog's three animations — 3₁3₁3₁, 3₂3₂3₂ and the rhombohedral r33₁3₂ — become exactly two after recolouring: the gyrational one, which is the colour symmetry of Escher's 1939 reptiles, and the sublattice one.

6. The circle: colouring without quantisation

Set aside the sampling. A forward spacetime group simply is a homomorphism-like cocycle φ: Γ → ℝ/ℤ — a colouring of the wallpaper group by the circle of phases — classified, as the notation page describes, by H¹(P, ℝ³/L) and its normaliser action. Every clockwork decoration is chromatic data read continuously: gyration subscripts are the colours of rotations, tildes the half-turn of colour on mirrors, the c and r prefixes the colours of translations, and the reversal clause the anti-unitary layer that discrete colour theory calls black–white. The clockwork orbifold symbol is, in this reading, a chromatic orbifold symbol for the colour group ℝ/ℤ — the discrete colour notations are its quotients, and the animation is what a colouring looks like when the colours are moments.

7. Directions

Four seem worth walking. (i) Coloured clockwork. A fully coloured clockwork orbifold — The Symmetries of Things' pair notation with clocks on the kernel, or equivalently Belov-style primes distributed over the bracketed projection's features — would unify this catalog with the two-colour tables and settle the residual superscript-letter collisions structurally. (ii) Colour × time. Animations of coloured patterns: cocycles into ℤ_M × ℝ/ℤ, whose classification would organise, e.g., animations that recolour as they loop. (iii) Non-cyclic colour groups. Perfect colourings with colour group S₃ or the quaternion trick of Wieting have no animation analogue yet — they would correspond to motifs carrying non-abelian internal states, a step from clocks toward connections. (iv) The chromatic count as homology. The tables above suggest reading the discrepancy between chromatic and animation equivalence as the action of the boost/re-basing group on H¹ — a finite computation this repository already performs, worth stating as theorems rather than counts.