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A magic theorem for clockwork animations

Five theorems about a periodic field whose spatial symmetries are allowed to cost time, their proofs, three worked examples, and a check against every animation this project has published. Nothing below assumes a differential equation, a bifurcation, or a small amplitude. Companion to the tutorial How to find a time-symmetric animation, which uses these statements but does not prove them.

What is being claimed#

Conway's Magic Theorem says that a wallpaper signature is legal exactly when its cost is 2. This report is the same shape of statement one level up. Decorate each symmetry of a wallpaper group with a fraction of a period — the time by which the animation must be shifted for that symmetry to hold — and a second closure identity appears, from the same orbifold relation, with the same proof in one line:

Conway's cost balances the geometry; this residue balances the choreography. It holds in all 68 entries of this project's catalogue, and — like Conway's — it enumerates: on the orbifold 333 there are 27 ways to hang offsets on the three cone classes, the identity keeps 9, relabelling collapses those to 4, and the catalogue's four p3 clockwork groups are exactly those four.

The identity does not stand alone. It is what makes two other forced structures compatible. A rotation centre whose generator costs a non-zero fraction of a period must be a zero of the first temporal harmonic — a point of the picture that does not oscillate at the fundamental frequency at all — and the phase must wind around it by an amount the offset fixes modulo the order of the centre (Theorem B). Those windings must cancel over one cell (Theorem C). B fixes their sum modulo 1 and C demands it be an integer, so B and C are simultaneously satisfiable exactly when the residue identity holds. That is Theorem D, and it is load-bearing rather than decorative.

Everything is a statement about one object: the first temporal harmonic of the field, one complex number per point of the plane. None of it mentions the equation that produced the animation, so none of it can be used up by changing the equation. What changes with the equation is only which of the permitted structures gets realised — and the last section of this report is about exactly that gap.

Shown at its own size here: drag the diagram sideways to read its labels.

Before any solver is started, the symbol alone decides a great deal about the picture: which points must be dark, how much phase must wind around each of them, and how the windings must balance across a cell. What follows is that statement, proved, and then tested against every animation the project has published. It is deliberately arranged the way Conway's Magic Theorem is arranged — a closure identity that the orbifold forces, and a topological budget that the identity makes satisfiable.

The data: a clockwork group#

A clockwork animation is a field $u(x, t)$ on the plane, periodic in time with period $T$, together with a wallpaper group $G$ and a map $\tau$ assigning to each symmetry a fraction of the period, such that

Apply the relation twice, once at $gx$ and once at $x$, and $\tau_{hg} = \tau_h + \tau_g$ drops out: $\tau$ is a homomorphism $G \to \mathbb{R}/\mathbb{Z}$, the modulo coming from $T$-periodicity. That single fact carries the whole theorem. Two consequences are worth naming at once. Its kernel $K = \ker\tau$ is the subgroup of instantaneous symmetries — the ones that hold frame by frame — and the quotient $G/K$ embeds in $\mathbb{R}/\mathbb{Z}$. It is not automatic that it is finite: a travelling wave $u(x, t) = f(k \cdot x - \omega t)$ satisfies the defining relation for every lattice translation with $\tau$ generically irrational, and then the image of $\tau$ is dense. Finiteness is an assumption here, and it is the one the catalogue encodes; for a group with a rotation it is also forced, because invariance of $\tau$ on the translations under an order-$n$ rotation clears the denominators. Granted finiteness, a finite subgroup of $\mathbb{R}/\mathbb{Z}$ is cyclic, so $G/K$ is cyclic of some order $m$.

This is the same data as a colour group whose colour permutation is cyclic: a wallpaper group with a homomorphism onto $\mathbb{Z}/m$, colours replaced by time offsets $k/m$. Every entry of the catalogue is listed in exactly that form on the correspondence pages — for the worked example below, correspondence-p3.html#g227, clockwork symbol r33₁3₂, orbifold 333, $m$ = 3. The catalogue's own notation page defines the decorations used below — the subscripts, the tilde for a half-period mirror, the prefixes c and r: notation.html. It is also the wallpaper instance of the pair $(H, K)$ of spatio-temporal and spatial symmetries that Golubitsky and Stewart attach to a periodic solution of an equivariant system, with $H = G$ and $H/K$ cyclic — here cyclic by construction rather than by hypothesis.

The object every statement below is about is the first temporal harmonic of a channel,

It is one complex number per point: the amplitude and phase with which that point oscillates at the fundamental frequency. Two pieces of housekeeping, once. A rotation centre in the plane is a gyration point; its image downstairs is a cone point of the orbifold, and "cone class" below means one orbit of gyration points. And index, winding and charge are the same integer throughout — the number of turns $\arg \hat A$ makes on a small loop — called by whichever of the three names the surrounding sentence wants. On a stored movie it is one bin of a fast Fourier transform down the frame axis, which is exact for the band-limited films the repository saves. Everything that follows is a statement about $\hat A$, not about the equation that produced it — no reaction–diffusion, no near-Hopf assumption, no small amplitude.

The worked example throughout is the catalogue entry g227: p3, orbifold 333, clockwork symbol r33₁3₂, $m$ = 3, and — as a colour group — the entry at correspondence-p3.html#g227. The decorations are defined on the catalogue's notation page.

Theorem A — the character law#

The sign is a minus. It is worth saying loudly because the plus sign is the natural guess and it is wrong: on the published g227 record the minus law holds to 8.6 × 10⁻¹⁶ — on a symmetrised export, of which more below — while the plus law is off by exactly 1.7320508 = $\sqrt3 = |1 - e^{4\pi i/3}|$, the largest error the character can produce. Flip any one of three conventions — the sense of the rotation, the sense of the winding, or the sign in the exponent of  — and the prediction flips with it. The repository's own earlier note left that sign as an unresolved $\mp$; this fixes it.

A corollary and an honest limit. Expanding $u$ in harmonics, the $j$-th term lives in the $\chi_j$ sector: the mean field is genuinely $G$-invariant and all the defects live in the sectors with $j$ not a multiple of $m$. But $\hat A_j \equiv 0$ always satisfies the constraint, so the theorem gives no lower bound — an animation may hide its structure in a harmonic that vanishes. In the 532 record–channel pairs tested, none did: $|\hat A_1|$ relative to the field's span stayed between 0.172 and 0.251.

Theorem B — what a cone point must contain#

The two halves are worth separating. Claim 1 says where the picture must be dark: any rotation centre whose generator carries a non-zero offset is a zero of the first harmonic — a point that does not oscillate at the fundamental frequency at all. Claim 3 says how the phase must run around it: the offset is a topological charge, read modulo the order of the centre. Neither statement mentions a partial differential equation.

Theorem C — the budget over one cell#

The lift-counting in claim 2 is the step that makes the budget an orbifold statement rather than a torus statement: a cone point of order $n$ is one point downstairs but $|P^{+}|/n$ points upstairs, so it enters the balance with weight $1/n$. The multiplicity law was checked against all 68 catalogue entries and is exact in every one.

Theorem D — the temporal magic theorem#

Conway's Magic Theorem says a wallpaper signature is legal exactly when its cost is 2 — equivalently, when the orbifold Euler characteristic vanishes. The time offsets obey their own closure identity, with the same source.

Theorems B and C are compatible if and only if D holds. B forces $\sum w_c/n_c \equiv -\sum k_c/n_c$ modulo 1; C demands that the same sum be an integer; the two can hold together exactly when $\sum k_c/n_c$ is an integer, which is D. That is the shape of Conway's theorem repeated one level up: Conway's cost balances the geometry; this residue balances the choreography.

What the residue means for a group with mirrors. The proof uses the orientation-preserving presentation, so for the forty-odd catalogue entries that have mirrors or glides the sum below is taken over the cone classes of the group's rotation subgroup, which is orientation-preserving and to which the theorem applies verbatim. That is what the residue column of the table records, and it is why a ∗-entry and its rotation subgroup share a residue.

Checked that way on all 68 catalogue entries, the residue is an integer in every one — the value is 0 for 27 entries, 1 for 32 and 2 for 9. But a consistency check is the least of it. Conway's theorem is famous because it enumerates, and so does this one. Take 333: there are $3^3$ = 27 ways to hang offsets on its three cone classes, Theorem D keeps the 9 with $k_1 + k_2 + k_3 \equiv 0 \pmod 3$, and relabelling the classes collapses those to four multisets — $\{0,0,0\}$, $\{1,1,1\}$, $\{2,2,2\}$ and $\{0,1,2\}$. The catalogue's p3 entries are 333, 3₁3₁3₁, 3₂3₂3₂ and r33₁3₂: exactly those four and no others.

What D does not do is guarantee existence. 4₁4₃2 has residue $\tfrac14 + \tfrac34 + 0$ = 1, an integer, and no group on a primitive lattice realises it; it exists only on a centred space–time lattice, as c4₁4₃2 — catalogue entry g99. The residue decides what may be decorated; the lattice decides what can be built.

Theorem E — mirrors#

Clause 2 is the familiar antisymmetry statement in a new setting: when $m = 2$ a clockwork group is a two-colour group, a half-period mirror is a primed mirror, and the statement is that the order parameter vanishes on a primed mirror. What is new here is that the same argument runs for every $m$ and every harmonic.

What this does and does not explain about the gaps in the atlas. Of the 51 catalogue entries with a non-zero offset, 35 have a Ginzburg–Landau record and 29 a Brusselator record. Every one of the 16 entries with no Ginzburg–Landau record, and every one of the 22 with no Brusselator record, is forced by the theorem to carry either a nodal line or a core of charge $\ge 2$ — 16/16 and 22/22, no counter-examples (Fisher exact, $p$ = 7.8 × 10⁻⁴ and 1.0 × 10⁻⁵). Three qualifications, all of which matter. First, that composite predicate was chosen after seeing which one reached 16/16 and 22/22: the scripts compute eight predicates against two models, sixteen tests in all, and the quoted $p$ values are not corrected for that choice. Second, a predicate with no theorem content at all — "the group has reflections" — reaches $p$ = 6.3 × 10⁻⁴ on the Brusselator gap, so the composite is not uniquely good; the sharpest single predictor is the forced nodal line alone, $p$ = 7.0 × 10⁻⁸, and that one the repository's own notes had already reported for Ginzburg–Landau. Third, pushing the other way, the search budget was uniform: the Brusselator batch gave exactly 4 jobs to each of the 51 entries, and all 88 jobs of the 22 entries without a record failed, which removes the most obvious confounder. The honest summary is a necessary condition satisfied with no exceptions, plus a plausible mechanism — a one-dimensional zero set is fragile under a perturbation that does not respect it, and a multiply charged core tends to split. It is not a demonstration that no orbit exists for the missing entries. The decisive experiment, running the solver on those entries and asking whether they fail for the predicted reason, has not been done.

Three worked examples#

g227 — r33₁3₂, the published spiral lattice

p3, orbifold 333, $m$ = 3, residue $0 + \tfrac13 + \tfrac23$ = 1, forced windings (0, −1, +1), and the budget $0 - \tfrac13 + \tfrac13 = 0$ closing on its own — so no free vortices are needed. Per p3 cell that is two charged centres plus one neutral one where $|\hat A|$ is instead maximal; on the simulation box, which is three p3 cells, six pinned vortices, three of charge −1 and three of charge +1, plus three neutral centres, total charge 0.

All of that is legible in the published animation. The pattern turns; the cores do not move, because Theorem B pins them to the group and not to the frame. Every third of a period the picture returns to itself rotated by 120° about a 3₁ centre — the plain centres are the ones that cost no time, and a turn about one of those is a symmetry of each frame on its own.

The pair of orbifolds is worth reading off, because it says what a simulator has to work with. In colour-group form g227 is 333/333: the kernel is again a p3, on a lattice three times coarser, so every frame separately still has threefold symmetry. Its neighbour g226 is 333/◦ — every rotation carries $\tau = \tfrac13$, the kernel is the translation lattice alone, and no single frame has any rotational symmetry at all. That is what the catalogue's $m$ column measures, and it is why the entries with a trivial kernel are the ones where a statement about the kernel-symmetric subspace has nothing to say.

g226 — 3₁3₁3₁, where the group offers a choice

Again p3 and 333, but now all three cone classes carry offset $\tfrac13$; the residue is $\tfrac13 + \tfrac13 + \tfrac13$ = 1, so the entry is legal. Theorem B allows $-1$ at each class, but then $\sum w/n = -1$, and Theorem C needs the total to be $0$. There are exactly two cheap ways out, and the group does not choose between them:

• move the leftover onto a cone — windings (−1, −1, +2), legitimate because $+2 \equiv -1 \pmod 3$, cost 4 cores per cell; or
• keep the minimal windings (−1, −1, −1) and pay for a free orbit of charge +1 — three extra vortices, cost 6.

Both branches occur in the published data. The Brusselator records and three of the four Gray–Scott rows take the doubled core; both Ginzburg–Landau records of g226 take the minimal windings with three free +1 vortices. The mirror-image statement holds for g225, whose Gray–Scott row also takes the free-vortex branch. It is tempting — and it was, at first, tempting here — to say that the doubled core is arithmetically forced. It is not: it is merely the cheaper of two admissible structures, and which one appears is a question for the dynamics, not for the symbol. The same caution applies to g244, where nothing is forced at all — its minimal structure 6₂[−2] 3₂[+1] 3₂[+1] already balances, and the one Gray–Scott record that departs from it simply chooses −2 at the threefold classes and pays with free charge.

g133 — 4₂∗~2, a mirror example

Orbifold 4*2, $m$ = 2. Its boundary is a single circle with one corner, so there is one class of mirror line — the two perpendicular families one sees in the plane are conjugate by the fourfold rotation — and it carries $\tau = \tfrac12$, which is the single tilde in the catalogue's symbol 4₂∗~2. By Theorem E.2 the first harmonic vanishes identically along that whole family of lines: the picture is a grid of dark lines, not a lattice of isolated cores. The geometry is worth getting the right way round, because it is easy to invert. The lines cross at the twofold centres, which therefore lie on a nodal curve and have no index at all — Theorem B's fourth case, and exactly what the measurements record. The fourfold centres, which carry $k$ = 2 = $n/2$, sit off the lines, at distance 0.354 in units of the cell edge, which is why they can carry the measurable charges ±2 that the budget needs. At such a chiral centre $+n/2$ and $-n/2$ are the same residue, so the group cannot fix the sign: this is the one place where the theorem yields a dichotomy rather than a value, and both branches — an isolated core of charge $\pm n/2$, or a crossing of $n$ nodal rays with no index — are realised in the catalogue.

What it buys, before any solver starts#

The statements above are not only a check on an answer; they are enough to draw one. Project a plane wave onto the character of Theorem A and most waves cancel identically — for g227, three of the six shortest reciprocal vectors survive and three read exactly zero, and the whole of the next shell cancels. Symmetrising one survivor gives a field with three Fourier components: three plane waves at 120°, which is the minimal p3 clockwork field.

That picture already has the structure Theorems B and C forced: exact zeros at the six screw centres with windings ∓1, maxima at the three $\tau = 0$ centres, total charge 0, and the character law to 7.0e−16. Lifting it through the oscillatory eigenvector of a 2 × 2 matrix and handing it to Newton's method on a twisted shooting equation reaches the published orbit in 5 steps. The construction, the solver and two live searches are in the companion tutorial, How to find a time-symmetric animation.

A structure the theorem permits is not a structure an equation will produce. Of the 51 catalogue entries with a non-zero offset, 29 have a Brusselator record; every one of the 22 without one is forced to carry a nodal line or a core of charge at least 2. That is an exceptionless necessary condition with a plausible mechanism — a one-dimensional zero set is fragile, a multiply charged core tends to split — and it is not a proof that no orbit exists. The qualifications are stated in full under Theorem E.

What is proved, what is measured, what is open#

Proved, from the defining relation alone, with no equation and no small-amplitude assumption: Theorems A, B, C, D, E. Each was independently re-derived during review. Measured, on unprojected data: the character law and the forced zeros and windings of g227, to 7.3 × 10⁻⁸ and 4.1 × 10⁻¹⁰ on two independent fields; and, on the symmetrised exports, the full catalogue sweep as a check of conventions and implementation. Open: whether a dynamically realised orbit can hide its structure in a vanishing harmonic — permitted by the theorem, never observed here; which branch of the half-offset tie a given equation selects; whether a forced structure is attainable at all, which is the stability question the coverage gap gestures at without settling; and harmonics beyond the fourth.