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.
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with Chaim Goodman-Strauss and Vladimir Bulatov, and the research it set off: a computer-checked reconstruction of an unpublished marked three-dimensional chair, the coupling notation behind the 184 composite space groups, an enumeration of the cyclic colourings that could sit on top of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.
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.
The equations, cyclic time constraints, domain length, and independent verification. The research note originally shared as the “note for Vladimir.”
Why the six forward-time characters are compatible with the equation, and what compatibility does and does not establish.
Parameter conditions that rule out certain periodic behavior before a numerical search.
Exact affine actions and the six compatible characters on the triangular lattice.
Required centres, extra symmetries of individual solutions, and apparent repeats that fail a strict check.
Connections and possible extensions of the spacetime-group project.
Saved-field checks of forward and inverse time shifts, and the coordinate conventions behind displayed rotation arrows.
Conventions, symmetry constraints, numerical methods, and the generator overlays.
Delivered data, orbit admission, parameter continuation, and spatial refinement.
Triangular-lattice dynamics, numerical evidence, and the limits of the original atlas.
Search, refinement, saved-data admission, and the shared catalog for the 17 galleries.
Ginzburg–Landau and Brusselator solutions with the same prescribed time-shift actions.
Pattern choices at fixed parameters, distinct spatial shapes, and the scope of the 442/632 Gray–Scott search.
Starting from known solutions, shooting constructions, and independent admission.
The original 442 search, GPU batching, measured CPU/GPU benchmarks, results, and costs.
Hopf seeds, correction, morphology searches, GPU batches, and reproducible branches.
Following periodic branches through parameter space and validating the saved outputs.
Mixed modes, deflation, secondary branches, localized seeds, and diversity criteria.
Recorded experiments extending the original square-lattice solution family.
Higher spatial wavelengths and why the prescribed time character still matters.
A reproducible comparison of the fundamental standing wave and its second spatial mode.
How saved orbits, catalogs, and verification evidence are organized.
The archived discussion and example animation associated with the original research note.