Research and development notes for Spacetime Groups.
Mathematical explanations, numerical evidence, and recorded experiments behind the animations. Each report states its own scope; earlier search counts describe the catalog at the time of writing.
How to go from a clockwork symmetry and a local equation to a looping animation. The whole method is first worked on three Brusselator cells on a ring — six numbers, a picture for every step, and a live widget — and then repeated on the g227 spiral lattice: why simulating forward fails, the seed you can draw with a pencil, Newton on one frame and one number, and a fundamental domain whose glued edges read its own past. Both searches run live in the page.
Notes from a meeting with Chaim Goodman-Strauss and Vladimir Bulatov, and the research it set off: a computer-checked reconstruction of an unpublished marked three-dimensional chair, the coupling notation behind the 184 composite space groups, an enumeration of the cyclic colourings that could sit on top of the 68 forward film groups, and three new three-colour animations.
What a clockwork symbol forces before any computing: the first temporal harmonic transforms by a character, each rotation centre carries a forced phase winding fixed by its time offset, the windings over a cell sum to zero, and the offsets themselves satisfy an integrality condition from Conway's orbifold relation. Checked on every catalogue entry.
Why a rotation, a time shift, and a color exchange can restore the same animation. A theory beginning with two colors and forward time, with the 442 family and the 68 polar animations as a starting point.
Seven weeks on one time axis: the owner's time, the agents' time and the Modal compute, day by day, with every artifact, the counting rules, and the history of the animations — what worked, what did not, and how hard each one was.