The making of animated-groups-fable

6. The animations#

Every animation in the Showcase is a solution of a reaction–diffusion equation that repeats in time and carries a clockwork symmetry: a spatial operation composed with a fixed fraction of the period returns the same film. The g227 spiral lattice is the clearest case. Its group is p3 with clockwork symbol r33₁3₂: turning the picture by a third of a turn about one class of centre and waiting a third of a period reproduces it exactly, about another class it is two thirds of a period, and about the third class no wait at all (correspondence-p3.html#g227). The concentrations obey the Brusselator equations pointwise, so no frame is drawn or forced; the symmetry is a property of one orbit of the flow, and the film loops because the orbit closes.

From the question to the orbit

28 and 31 August — the question

The reaction–diffusion arm began as a prerequisite, not a plan. In an email to Vladimir Bulatov late on 28 August the owner asked whether Vladimir's differential-equation solver could be modified to produce periodic animations. Three days later, on 31 August at 10:19, he wrote out the research question itself in another email: wallpaper symmetry applied to differential equations is familiar, but could it be extended to time symmetry — a looping animation whose right half is a mirrored copy of the left, offset in time. The stated destination was a poster for the next Gathering for Gardner conference, in San Francisco. Nothing was built for four days.

4 September — Gray–Scott, one bug, and the project's only GPU afternoon

At 13:54 the owner opened a session with that same question pasted in, named clockwork symmetry 442 and its six forward-time characters, and attached two screenshots of U-skate gliders as the visual target. The first two minutes went on algebra rather than code, and it paid: time reversal is obstructed for an autonomous isotropic flow, because differentiating the reversal relation forces the orbit to be stationary; and a 442 pattern cannot be a rigidly drifting glider lattice, because a quarter turn sends the drift velocity to Rc and a common drift needs Rc = c. All six characters were shown compatible; the glider picture was excluded at the level of the group on day one.

The first solver was a space–time collocation: Crank–Nicolson defects over M frames on an N × N torus, symmetry imposed as an exact projection, minimised by L-BFGS with a hand-derived adjoint gradient. Six U-skate searches at 200 iterations each drove the residual down and held the time symmetry exactly — and every one failed independent forward closure. The defect on a coarse frame stack does not control the true trajectory.

Then the correction that shaped the whole codebase. At 14:40 the owner checked a few of the live animations and reported that they don't seem to respect the time symmetry, adding that switching between g95 and g96 made no difference. The diagnosis came two minutes later: the browser integrator had been projecting each step onto the instantaneous kernel — the operations with zero time shift — and then drawing the requested phase-shift generators on top. For g96 that kernel is trivial and for g95 it is a half turn, so the two rendered identically and only the decoration changed. A time-shift character is not a state-space constraint and cannot be imposed step by step.

The fix was to stop constraining the flow and constrain a return instead. Solve $\Phi(T/m,\ q_0) = q_0 \circ g^{-1}$ for an initial state and a period, with $\Phi$ the ordinary unprojected evolution, the instantaneous kernel reducing only the unknowns, and an analytic Hopf eigenmode as the seed. By 14:48 a rotating g96 wave had an independent return error near $10^{-12}$; by 14:55 both branches — one standing, one rotating — were in hand, and the six 442 characters were populated by translations and reflections of those two. An earlier decision made that verification load-bearing: asked at 14:24 whether a working animation meant one tested to move or one verified periodic, the owner chose Only verified periodic solutions, even if the list is empty.

Modal entered at 14:47 with a smoke test for A100 reservations, and the search budget was capped at $100. Every GPU-second the project ever bought falls in the 2½ hours from 14:59 to 17:30 that afternoon: 2.398 A100-40GB-hours, $5.64 metered on the day, all of it Gray–Scott. The kernel was FP64 with fused multiply-add disabled so that GPU and CPU agreed bit for bit; on a 1,252-trajectory batched Jacobian it ran 148.9× and 149.6× faster than the same C++ integrator on one core. One prototype tried to return 66 MB of movies from the cloud and stalled after the computation had already finished; the redesign returned about 0.5 MB of initial states for 14 orbits and rebuilt the films locally, which also kept the reconstruction independent.

The triangular 632 extension the same evening cost nothing at all. Because a rotation permutes Fourier modes of equal Laplacian eigenvalue, the twisted linear flow factorises into 2 × 2 blocks; a Fourier-block preconditioned Newton–Krylov solve took 2.3 s where dense Newton on the same grid took 222 s, and 22 verified orbits covering all six 632 characters were computed locally with no cloud jobs. What did not work that evening was variety. At 16:05 the owner reported that every setting was producing the same pattern and asked for twelve to twenty per parameter set; the search delivered 8 in 442 and 11 in 632 from 309 jobs, because the symmetry reductions that made the shooting fast were also funnelling every seed into one branch family.

5 September — all seventeen wallpaper groups

Now do this for all the 17 symmetry groups, at 21:57. Twenty-four minutes later a commit added 405 files and 100,714 lines: a general solver reading canonical affine operations for all 68 forward-time entries, 441 bounded CPU jobs, all returned exactly once. The raw shooting success rate at the exploratory stage was 77 of 328 — 23.5 % — refinement to a finer grid succeeded 76 of 77, and end-to-end admission was 22.6 %. One character resisted: a p6m mirror needed a dedicated 36-job campaign and yielded 12 convergences of which only 5 cleared the visibility floor, because small-amplitude mirror solutions can be mathematically correct and invisible. The whole day's cloud bill was $0.125.

Earlier the same day the work turned outward. Video exports were cut for sharing, the session transcript was published on the site, and a research note was written explaining what had been done to the equations — nothing, as it turned out: the equations are unchanged and the additions are the two relations $q(x,\ t+T) = q(x, t)$ and $q(gx,\ t + \tau_g T) = q(x, t)$, with no extra reaction term, forcing or time reversal. Vladimir's comments were relayed into the session three times that afternoon and evening. He was right about the overlays: they had repeated only three named representatives by integer cell translations, which generates the group but under-draws its rotation centres. The audit that followed checked 1,500 derived rotation-and-phase operations at 3,448 centres on every frame and node of both species, largest error $2.98 \times 10^{-8}$, and separated the group's required centres from extra centres that only this solution has and from approximate near-periods that are never certified.

6–7 September — two more equations, and the spiral lattice

On 6 September at 16:34 the review session opened on the same unsolved complaint — insufficient diversity — with a pointer to Vladimir's Ginzburg–Landau app and a request for several equations behind the existing seventeen pages. Ginzburg–Landau turned out to be nearly free. The equation is equivariant under a global phase rotation, so a state of the form $B(x)\cdot e^{i\Omega t}$ is periodic by construction and a spatial phase winding of $B$ is a time shift; the constrained subspace is flow-invariant, so plain integration inside it plus a Newton–Krylov polish to residual near $10^{-13}$ gives exact spiral lattices. It covered 35 of the 51 entries with a nonzero time shift, and the 16 misses are principled: an operation that is a mirror or glide with a half-period shift forces the amplitude to vanish along a line, and those line defects are unstable, so the unprojected replay rejects them.

The Brusselator was reached through that result rather than around it. Near its Hopf point the Brusselator reduces to a Ginzburg–Landau equation, so the job builds a matched spiral lattice on the same mesh with a rescaled cell, lifts it through the Hopf eigenvector of the reaction Jacobian, and closes the loop with Newton–Krylov twisted shooting — $S_g \cdot \Phi_{T/3}(q) - q = 0$ on the unprojected flow, with $\log T$ among the unknowns. The batch bruss1 launched at 11:00:48 on 7 September: 204 CPU jobs, no GPU, 81 candidates and 122 failures, 4,654.52 s of function time in total, and an actual provider charge of $0.2007. The orbit on this page came back from job g227-bruss-a1b23-L40-s1 after 53.75 s of function time, somewhere inside the eleven minutes from 11:00:52 to 11:11:56 — Modal records no per-job start time, so the moment of discovery is bounded, not pinned. Its share of the bill is about $0.0023. It was published at 12:30 inside the review's single commit.

The recorded negative result of the same session is worth as much as the positive one: plain forward integration with kernel projection — the strategy of the reference implementation — was tried for the Brusselator and for the Barkley excitable model and does not work for time-shift characters. It synchronises to the uniform oscillation or drifts. This is the 4 September bug restated as a general statement.

The afternoon of 7 September added two things that outlived the era. A handedness audit, prompted by the owner reporting that the generator glyphs looked backwards, found that the artwork transform had omitted the display camera's vertical reflection, and that identifying conjugate centres by generator name was wrong because a spatial reflection conjugates a rotation to its inverse while keeping the offset. The catalogue-wide survey then checked 927 records and 5,598 operation–record relations with zero wrong-sign passes: the films had not been running backwards, the arrows had. And Ember, the first standalone viewer, shipped at 15:04: since a Ginzburg–Landau orbit is a relative equilibrium, the entire film is one 10,368-byte texture turned analytically, with no frame stack at all.

16–21 September — the viewers, the catalogues and the Showcase

After eight dormant days the work restarted on the second Mac, in a different direction: not finding orbits but looking at them. The commits from 16 to 20 September — 24 of them — all carry the trailer Claude Fable 5.1. The first ask, at 10:10 on 16 September, was only about resolution; everything that makes Plume Plume arrived as four follow-ups over 21 hours — opening it on the phone, one finger to pan, two fingers to turn, then Safari. The rewrite at 11:13 blended frames on the CPU into one float texture and reconstructed on the GPU with nine bilinear taps instead of the previous per-pixel cost; the displayed value differs from the exact band-limited reconstruction of the saved samples by under 0.4 of one 8-bit colour level, against up to 9 levels on the catalogue page it was made from. Plume Monochrome was added the same morning as a byte-identical twin that differs by one attribute on the canvas.

Two records disagree about who built Plume Monochrome, and both are given here as they stand: an email sent at 11:53 that morning credits the Codex model gpt-6-astra, while the two commits of that hour carry Co-Authored-By: Claude Fable 5.1 and no Codex session touched the repository in the window.

17 September produced a whole three-colour family in an afternoon. One prompt at 08:08 opened a workflow that fanned out to 55 subagents; a request at 10:37 for an animation whose symmetry needs a spatial generator, a time shift and a recolouring together produced Gyre, and Triskele alongside it as a control that nobody asked for, to make the contrast legible. At 13:36 came the sharper case, a colour permutation that is not cyclic, and that is Trefoil. All three paint the same 3.35 MB Gray–Scott field, byte for byte; only the colour rule differs, and the rule decides the colour group. That is the point of the set, and it has a proof behind it: composing two symmetries adds their time shifts, so the time shift is a homomorphism to a circle, and a colour group can be totally entangled only when it is cyclic.

18–19 September is the densest stretch of the project: two prompts on the evening of the 18th, then Weave Monochrome at 00:00, the colour catalogue at 02:41, the finished equation search merged at 04:15 — all three committed with nobody in the loop, the next prompt arriving at 07:47. The colour search that produced the catalogue ran locally: 14 workers against a four-hour deadline, 3,106 of 8,502 planned jobs, 770 admitted records, rebuilt into 423 exact colourings — 346 Gyre and 77 Trefoil — over nine equations. The monochrome search went to the cloud at 08:12: 2,905 jobs submitted and returned, 849 orbits admitted across all 68 film groups, actual charge $3.808 against a self-imposed worst-case reservation of $69.91 the agent wrote into its own ledger. Its lesson is in the split: the relative-equilibrium strategy admitted about 40 %, the Hopf-seeded shooting strategy about 5 %, and two of the zeroes are structural rather than unlucky — Gray–Scott cannot be found by the twisted equation at all, because a symmetric steady state satisfies it at every period and Newton drives the period to zero, and three of the models simply synchronise to their uniform limit cycle. The biggest disappointment is also recorded: on the square lattice 802 of the 849 admitted records are single-Fourier-mode travelling stripe waves. Genuine orbits, one motif.

Crosslet shipped at 09:21 the same morning, the one viewer with no Cloudflare project because the prompt asked for GitHub Pages; and at 14:35 a single commit of 7,613 files added the Monochrome explorer, the site-wide navigation on 88 pages, and the Showcase: every precomputed animation on one page, 4,138 cards over 2,632 preview clips. Making that page readable took two more days and two attempts. A clip metric — 36 frames at 40 × 40, blurred and z-scored, difference minimised over cyclic shifts in time and both screen axes — folded 2,153 of the 4,138 cards. It was not enough: six cards of one rotating wave, read under six different symmetries at six feed values, cycle through the same shapes in a different order, so no alignment of one loop onto another is close and the metric called them unrelated, while a reader calls them one card six times. A second fold by name brought the page to 1,070 distinct-looking cards with 3,068 folded.

29 September — two standalone pages, and the question that came back

Six prompts between 10:06 and 11:29, 18.1 minutes of agent-busy time, five commits, and the g227 field left untouched: the two Cloudflare Workers serve the same 497,664 bytes as the catalogue, byte for byte. The colour page went up at 10:08, the monochrome twin at 10:18, both were then rebuilt on Plume's fullscreen viewer, and the monochrome cut was raised from zero to 0.426 — 30.1 % of the maximum — because at zero the two inks trade blob and network roles six times a period, while at this level the white islands neither merge nor split, checked at 768 instants. The price is stated on the page: the exact half-period colour swap is given up.

At the symmetry meeting that morning Vladimir asked how the periodic solution is actually found, and whether it resembles solving a stationary Schrödinger equation — an eigenvalue problem for the period. Another participant asked whether all the frames are computed together and evolved at once, in effect in a fifth dimension. Both questions have answers in the record. The eigenvalue framing is right for one of the three equations and only partly right for the other two: for Ginzburg–Landau the code really does solve $F(B) = i\Omega B$ with $\Omega$ as the unknown frequency, whereas for Gray–Scott and the Brusselator the problem is a fixed point of a twisted return map and eigenvalues enter through the seed and the preconditioner. And computing all the frames together and relaxing them is exactly what the first solver did on 4 September — the space–time collocation — which is the one that failed. The owner's own account at the meeting was that his contribution was to specify the problem and that he did not know how the looping animations had been found; the answer was promised before the next meeting.

What worked, what didn't

Every approach that was tried on the animation problem, with its result and the cause. Four entries never reached code at all; they are listed because they were designed and are still the named next steps.

How hard was it

Per family, in the units each was measured in. Agent-hours are measured turn intervals summed across parallel threads, so two agents busy for an hour count two; dollars are Modal's metered charge for that family's apps, not the agents' worst-case reservations, which over-reserved by ten to twenty times. Corrections counts prompts in which the owner said something was wrong or missing, excluding new requests and quota notices. The difficulty rating is the one recorded in the research files, on a scale of five.

The asymmetry worth naming: the orbit that carried the project into its last meeting took 53.75 s of one CPU container and about 0.2 of a cent. Everything expensive happened before it — the failed collocation solver, the projection bug, the twisted return equation, the GPU afternoon, the seventeen-group generalisation, and the amplitude-equation route that let a Ginzburg–Landau spiral seed a Brusselator one. The search was cheap because the method was finished; the method took four days.

The viewers, one by one

Eleven standalone pages were built between 7 and 29 September, ten of them from a single orbit each and the eleventh from a tiling. The model named in each card is the one in the commit trailers; untrailered commits are Codex.

Ember

7 Sep 15:04 · Codex (gpt-6-astra)

Equation and group

Complex Ginzburg–Landau; g134 = P4₂nm, family p4g, signature ²4*¹2; 36 × 36, two channels.

What was hard

Nothing, and that is the result: because the orbit is a relative equilibrium, the whole film is one 10,368-byte texture rotated analytically, with no frame stack and no interpolation in time. The trick is unavailable for Gray–Scott, which is why Plume needed a 2.36 MB field nine days later.

Live

p4g-ember.pages.dev

Plume

16 Sep 10:26, rewritten 11:13 · Claude Fable 5.1

Equation and group

Gray–Scott $F$ 0.00395, $k$ 0.02; shown on the 2222 page as g6 = P2₁, while the field's own entry is g96 = P4₁ (442, ⁴4⁴4²2); 128 frames × 48 × 48.

Clockwork

A quarter turn is a quarter period: $U(Rx,\ t + T/4) = U(x, t)$, exact on the saved samples.

What was hard

Interaction and GPU numerics. The first shader cost about four times too much per pixel; the exporter built its own renderer and produced the monochrome twin in the wrong palette; the shader turned the opposite way from the view model, caught only by a browser test driving a simultaneous turn and zoom; Safari reports trackpad pinches as gesture events, not pointers. The payoff is a displayed error under 0.4 of one 8-bit level.

Live

plume-wave.pages.dev

Plume Monochrome

16 Sep 10:32 · Claude Fable 5.1 by the commit trailers; credited to gpt-6-astra in an email the same morning

Rule

$\operatorname{sign}\bigl(U(x,t) - U(-x,t)\bigr)$, zero contour anti-aliased over one device pixel.

Colour group

The quarter turn with a quarter-period shift preserves the two colours; the half turn, the half-period shift and the quarter turn with $3T/4$ reverse them — a two-colour spacetime group, exact on the samples.

What was hard

Almost nothing: the code, the renderer and the field are byte-identical copies of Plume's and only the page differs. The limitation was learned later — a film group hands you one monochrome picture per orbit however many half-period involutions it has, which is why the later search had to mine free involutions instead.

Live

plume-monochrome.pages.dev

Weave Monochrome

19 Sep 00:00 · Claude Fable 5.1

Equation and group

Gray–Scott $F$ 0.00395, $k$ 0.02; “woven rotating wave · (3,1)”, catalogued as g11 = Pc (pg, ××), 48 × 48 × 128, square cell $L = 256$, $T = 352.0373$, 96 % of the power in one reciprocal star.

What was hard

Measurement, not code. An exhaustive search finds 32 exact symmetries of the samples — mirrors and glides the catalogue's operation list does not mention. Three rules that sound different are bit-identical on this orbit, and the alternative rule turns out to be the same twill at another instant, agreeing 99.4 % at a lag of 51 frames. The picture is exactly half black by force. Cloudflare will not compress application/octet-stream, so the field ships pre-gzipped with a loader that sniffs the magic number.

Live

weave-monochrome.pages.dev

Triskele

17 Sep 14:49 · Claude Fable 5.1 · not requested

Equation and group

Gray–Scott $F$ 0.00406, $k$ 0.02, on the shared 66 × 66 × 96 triangular field, $L = 512$, $T = 333.9869$ so that $T/3$ is exactly 32 frames; field groups p3 g225 (333, ³3³3³3) and p6 g247 (632, ⁶6³3²2).

Colour group

$\mathbb{Z}_3$, split: the turn alone recolours, the $T/3$ wait alone recolours, and doing both changes nothing — the control case.

What was hard

Little; the rule is the cleanest of the three. An exhaustive search over every point operation, translation, saved time shift and both time directions finds exactly the six turns of the field and nothing else. Half the download is fetched and discarded to keep the shipped file byte-identical to the atlas file, and the frame shutter was later cut to 0.3 of a frame for a crisper picture.

Live

triskele-wave.pages.dev

Gyre

17 Sep 14:09 · Claude Fable 5.1

Rule and group

The same field as Triskele, byte for byte, coloured by $\operatorname{argmax}_k U(g^k x,\ t + kT/3)$ about a point that is not a symmetry centre — 0.0962 lattice lengths from the nearest threefold centre, chosen so the displacement is exactly 11 of 66 nodes and every claim is integer arithmetic. Colour group $\mathbb{Z}_3$, fully entangled.

What was hard

The conceptual centrepiece: 1,616 lines of viewer and 2,043 of tests. The two halves of the law reach at best 0.4284 on their own — barely above chance — which is what makes the entanglement visible. The honest qualification is recorded too: because this field carries the threefold screw, the three “different instants” collapse to three readings of one frame, and the mining report finds the same collapse for 123 of the 200 catalogued Gyre colourings.

Live

gyre-wave.pages.dev

Trefoil

17 Sep 16:03, generator marks 20:28 · Claude Fable 5.1

Rule and group

The same field again, read under g247 = P6₅, symbol 6₅3₂2₁, coloured by three positions at one instant — a slide by the step between classes of threefold centres, exactly (22, 44) nodes. Colour group $S_3$, not cyclic: a slide cycles the colours with no time shift, while a half turn plus half a period swaps two and fixes the third.

What was hard

The hardest of the set: 2,387 lines of viewer, 2,924 of tests, and a README containing an original theorem — in any three-coloured polar film with colour group $S_3$ the 3-cycles are always spatial and the swaps are all-or-nothing. Two independent exhaustive sweeps over 30,108,672 combinations agree on exactly 18 symmetries, and every drawn generator is exact on all 418,176 node-frames. No mirror survives anywhere; the square-lattice route to $S_3$ had already been rejected because the argmax ties along whole lines.

Live

trefoil-wave.pages.dev

Crosslet

19 Sep 09:21 · Claude Fable 5.1 · GitHub Pages only

Equation and group

Complex Ginzburg–Landau, $\alpha$ −0.75, $\beta$ 0.55, $D$ 1; the catalogue entry colour:gyre:a767fbcc68c0 on p3 g225 (333), 36 × 36 × 96, triangular $L = 64$, $T = 11.3164$; Gyre's rule about a point a third of the way across the cell.

What was hard

The near misses are near. The law is exact at all 124,416 node-frames and an exhaustive sweep over 8,957,952 combinations finds exactly three symmetries — but the two halves of the law reach 92.9 % on their own and the best mirror near miss 93.97 %, so the test has to be exactness and never agreement. Every node changes colour exactly three times a period; the boundaries move nearly three times faster than Gyre's.

Live

…/scott-gray/crosslet/

Spiral Lattice

29 Sep 10:08, rebuilt 11:32 · Claude Code, then Claude Opus 5.5

Equation and group

Brusselator $a = 1$, $b = 2.3$, $D_u = D_v = 1$, triangular six-neighbour stencil, $N = 36$, $M = 48$, $L = 40$, $T = 6.3902$; g227 = R3 (No. 146), p3 / 333, clockwork symbol r33₁3₂, signature ³3³3¹3.

What was hard

Nothing in the mathematics — the field is the catalogue's own bytes — but the viewer had to be ported to a triangular lattice: the same fullscreen WebGL renderer with a lattice map, turns snapping to sixths, and a threefold centre on screen at 380 pixels per lattice length, one period per 8 s.

Live

brusselator-p3-spiral.mathornament.workers.dev

Spiral Lattice Monochrome

29 Sep 10:18, threshold raised 10:30 · Claude Code, then Claude Opus 5.5

Rule

$U(x,t) - U(x,\ t + T/2) > 0.426$ — the catalogue's half-period rule, cut at 30.1 % of the maximum instead of zero.

What was hard

The trade-off, which is the day's one new result. At a cut of zero the two inks trade blob and network roles six times a period; at this level the white islands neither merge nor split, verified at 768 instants — and the exact half-period colour swap is given up to get it. The page says so.

Live

brusselator-p3-spiral-mono.mathornament.workers.dev

Chair tiling

17 Sep 14:09 · Claude Fable 5.1 · not a reaction–diffusion animation

What it is

A WebGL view of the marked three-dimensional chair substitution described at the meeting that morning, reconstructed from a sketch and checked by program: substitution consistency at levels one to three, exterior self-similarity, and complementarity of the dents and backs. The verifier ships with it.

Live

chair-tiling.pages.dev