Colour
Entangled colour symmetries
All three pages below paint the same verified reaction–diffusion waves — and, from this release, a whole catalog of them, over nine reaction–diffusion equations and every hexagonal film group the catalog found one in (the counts are on the right, and they grow as the search runs). Only the rule that assigns the three colours differs, and that rule decides the colour group: whether a recolouring can be seen in a single frozen frame, or only in the film.
Composing two symmetries adds their time shifts, so the time shift is a homomorphism to a circle. Every recolouring that needs no time shift forms a subgroup ker θ̄, and the quotient Π / ker θ̄ — a finite subgroup of a circle — is therefore cyclic.
So a colour group can be totally entangled only when it is cyclic: Gyre, with Z₃, gets away with it. A non-cyclic colour group such as S₃ must leave part of itself visible in a single frozen frame — Trefoil, whose 3-cycle is a plain slide and whose swaps are welded to a half-turn and half a period.
Triskele is the control: the turn alone and the wait alone each recolour, so the entanglement splits and every frame is a threefold rosette.
Three rules, three colour groups
| Page | Colour group Π | ker θ̄ — recolourings needing no wait | What one frozen frame shows |
|---|---|---|---|
| Gyre | Z₃, cyclic | trivial where the entanglement is total | on a fully entangled entry, nothing but the lattice: no rotation centre, no mirror, no colour symmetry at all |
| Trefoil | S₃, not cyclic | the 3-cycle — a plain slide by b | a threefold colour cycle on the finer lattice; the two-colour swaps are invisible |
| Triskele | Z₃, split | the whole of Π | a threefold rosette, fully coloured — the turn alone already recolours |
The rule is the same on every entry; the third and fourth columns describe the character this catalog selects for, and the figures beside them count the entries that actually have it. Each explorer page says, entry by entry, which side of that line the picture on screen falls.
How to read the marks
The coin — what the motion is and how long you wait. The site's rotation-order glyph with its screw tails, a violet clock dial filling k/n clockwise from twelve, and a short anticlockwise arc for the sense. An empty ring means no waiting at all.
The chip — what becomes of the colours. Three palette dots at fixed stations: 0 terracotta at twelve, 1 teal at four, 2 sand at eight. Chasing arrows cycle all three; a double-headed arrow joins the two it exchanges and rings the one left alone; three plain dots mean the colours never move.
The plain arrow — a slide, with no turn and no wait at all. A full head means a whole step of the picture's own lattice. Written down, the two decorations are the symbol nk(σ): see the notation page.
Only entries whose colour law was re-derived from the saved float32 bytes at every node-frame — with zero violations and zero argmax ties — appear here. The orbits themselves come with two different warrants, and each entry prints its own: an atlas orbit carries the wallpaper atlas’s offline verification, and a search record carries a re-integration from the exported bytes at two timestep limits with no symmetry projection. 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, and every hit is re-checked in exact integer arithmetic. Fields that cannot host a colouring are listed, with their reason, on each page.