Trefoil slide freely, swap only with a half period
Three colours assigned by which of a point’s three translates on the finer lattice L′ = L + ℤb leads, all at the same instant. Sliding by b cycles the three colours with no waiting at all; exchanging two of them needs a half-turn and half a period, and no combination of the lattice can do it without one. The colour group is S₃, which is not cyclic — and a non-cyclic colour group must leave part of itself visible in a single frozen frame. Every entry below is a saved orbit carrying its own warrant — the wallpaper atlas’s offline verification, or a re-integration from the exported bytes at two timestep limits with no symmetry projection; each entry prints which — and its six recolourings were confirmed by exhaustive search in integer arithmetic on the samples.
1 · The base film group
This colouring needs something of the field: an exact symmetry h with h b ≡ −b (mod L), which reverses the compared list and so realises a swap. A half-turn always supplies one; some mirror groups do too — a mirror with h b ≡ −b would give a swap with no time shift at all — but no such field survives the tie test, so every admitted entry uses the half-turn. And only where the argmax never ties — on a standing wave b is nearly a period of the field and the three compared numbers are equal almost everywhere. Every admitted field in the catalog uses R180 at half a period, so every one of them has entangled swaps — the catalog offers no counterexample.
The colour law is re-derived from the saved bytes at every node-frame before an entry is published, and the colour-preserving subgroup is measured by an exhaustive search over all twelve point operations, every node translation, every time shift and all six recolourings — not assumed.
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Parameters, thumbnails and the measured symmetry tables are precomputed.
Coin: the turn and how long you wait. Chip: what becomes of the colours.
Measured symmetry table
| Operation | Time | Colour | Agreement |
|---|
Numerical record
Only colourings whose law was re-derived from the saved bytes at every node-frame appear here.
Why some fields cannot host this colouring —
Every refusal is logged with its reason in colour-atlas-build.json, beside the catalog. Some refused fields do carry a mirror with h b ≡ −b — which would give a swap with no time shift at all, and put the film in the other branch of the theorem — but every one of them is a standing or breathing wave for which b is nearly a period, so the argmax ties and the colour group collapses to ℤ₂ or to nothing. The catalog therefore offers no counterexample; finding one needs a purpose-built field, even in x at every instant.
Entangled colourings · Gyre · Z₃ · Triskele (the split sibling) · The standalone Trefoil page · Notation · How the patterns are computed