Research report · 7 September 2026

Space, time, and color symmetry

A movie can return to itself when we rotate the plane, advance its phase, and exchange its colors together. The right object is the group of these triples. A particularly simple theorem explains why a half-period color exchange produces a whole second family of symmetries.

1. What the observed movie actually does

The motivating 2222/g6 animation uses the saved field g96-F0p00395000-k0p02000000-N48-M128, with period T ≈ 398.977432. Its catalog provenance is audited-subgroup: the field was tested against the smaller group, with extra symmetry allowed. Thus its 2222 label describes a verified subgroup, not necessarily its full symmetry.

Write q = (U,V) for the two concentrations and let R(x,y) = (−y,x), a counterclockwise quarter-turn about the origin in mathematical coordinates. In units of one period, the data satisfy

The Ember palette is a multicolor gradient of U alone. Its colors are neither the two chemical species nor literal two-color labels. To make “change the colors” testable, we audited two concrete interpretations: reversing the normalized scalar palette coordinate, and swapping two labels obtained by thresholding U.

RMS averages every stored node and frame, normalized by the full movie's max−min for that channel. The threshold test counts unequal labels after exchange. These tests do not exhaust arbitrary nonlinear color maps, other rotation centers, or every possible phase. Byte-exact identities on this aligned grid also survive periodic linear/bilinear playback interpolation; they are not a proof of an exact continuum PDE solution. Full audit · Reproduction script.

2. Give each symmetry three coordinates

For a two-color movie, encode the labels by f(x,t) ∈ {−1,+1}. A signed scalar can generate these regions: its sign gives the labels, with its zero set treated as an uncolored boundary. Requiring the same identity for scalar amplitudes is a stronger test than matching just the labels. Normalize the movie's primitive period to 1. Define

These triples form a subgroup of E(2) × (ℝ/ℤ) × C₂. Composition is

A symmetry may therefore use all three coordinates even if none of its parts separately preserves the movie. For k colors, replace C₂ by the permutation group S_k and replace the sign by a single global permutation of labels. This is a color-symmetry extension of ordinary space–time symmetry, not a redefinition of chemical concentrations. See the established color-group framework.

3. The half-period theorem

Proof. A pure phase shift with no color exchange is a period, hence zero modulo the primitive period. Squaring a pure phase-plus-swap gives a pure shift by 2τ, so 2τ = 0 modulo 1. The choice τ = 0 would say the movie equals its own color exchange; the only remaining choice is ½. Two different swap offsets would differ by an ordinary period, so there is at most one. Composing with this kernel element gives the companion. Two lifts of the same spatial operation differ by a kernel element, so there are no others.

Applied to 442. Suppose a quarter-turn already gives (R,¼,0). If the movie also has the half-period swap (e,½,1), composition forces

This directly answers the “another level” observation: one additional temporal/color symmetry doubles the number of lifts of each spatial operation. It need not add a new spatial operation. Separately, a particular movie can also have a larger spatial projection, as the saved g6 example does.

The character form. Let G be the spatial projection of S. If N is trivial, the unique lift defines a homomorphism G → (ℝ/ℤ) × C₂. If the half-swap is present, lifts instead define a homomorphism into the quotient by N. This quotient is a circle: the invariant coordinate is θ = τ + ε/2 modulo 1. Thus color and time can become two descriptions of the same effective phase. This precise distinction prevents counting paired lifts as independent new group types.

4. An exact two-color movie with 442 symmetry

The following analytic model has the same directed quarter-turn relation as the saved g96 field. It realizes the required 442 subgroup and also has additional spatial symmetries; its full symmetry is not asserted to be only 442. It is a mathematical illustration, not a reaction–diffusion solution. On the unit square torus, set

Direct substitution gives f(Rx,t+¼)=f(x,t) and f(x,t+½)=−f(x,t). Therefore f(Rx,t+¾)=−f(x,t). The three panels below show the original, the spatial/temporal transform, and the result after the selected color operation.

Increase the static component to add ε[cos(2πx)+cos(2πy)]. It is unchanged by R, so the quarter-turn/quarter-period symmetry survives. But it does not change sign after half a period, so the color-swap symmetry breaks. The demo checks 12,696 samples spread across the whole loop, rather than trusting one matching frame. The formula proves the exact identities; the numerical check is a diagnostic.

At nodal points a literal two-label assignment cannot equal its own swapped label. The boundary is therefore excluded from the colored regions; sampled zeros are drawn in the page's neutral color. In the image comparison, f(Rx,…) is a pullback: visible features can move in the opposite direction to the coordinate rotation.

5. Describe 442 using generator triples

The orientation-preserving wallpaper group p4 has orbifold signature 442 and presentation

When lifts are unique, decorate each generator by its phase and color bit: a ↦ (τ_a,ε_a), and similarly for b,c. The defining relations impose

The order relations give no further restriction on a two-color bit because 4 and 2 are even. Thus there are four color-character assignments for these named generators:

These are algebraically consistent homomorphisms, not four new inequivalent animation types or a promise that every motif realizes them. Equivalent choices of origin, generator names, and lattice can identify decorations. Motif stabilizers impose additional constraints: a color-swapping symmetry with zero phase cannot fix a point in the interior of a colored region.

If the half-period swap is present, use the quotient character θ instead of treating τ and ε as independent. Generator relations must hold in that quotient; a relation's lift can end at the nontrivial element of N. In particular, for a spatial operation of order m, the general rule is (e,mτ,mε) ∈ N. If N is trivial this is the usual pair of zero congruences. This also explains a useful parity check: an odd-order spatial rotation cannot exchange two colors at zero phase, but it can participate in a phase/color operation whose odd power is the half-period swap.

There is no need for an ambiguous prime on “442.” Record a presentation, the concrete centers/matrices, and the triples. That describes space, phase, and color simultaneously and makes every assertion testable.

6. Why these symmetries arise

A single oscillatory mode has an automatic sign reversal. For f(x,t)=Re[A(x)e^2πit], advancing half a period multiplies f by −1. If A transforms by a one-dimensional complex character under a spatial operation, its phase can be canceled by a time shift. These two facts together create the three-part symmetry. This is the representation/character construction used in the spatio-temporal point-group literature.

The exact Fourier test is stronger than a visual resemblance. Expand a signed, periodic scalar field as f(x,t)=Σ_n f_n(x)e^2πint. A triple is a symmetry if and only if every Fourier coefficient obeys

Near an oscillatory bifurcation, the symmetry can be approximate. A leading sinusoidal mode changes sign after half a cycle, while nonlinear terms generally introduce a stationary correction and higher harmonics. An underlying color/sign equivariance can prohibit the unwanted terms; without it, there is no reason for exact cancellation. For a concentration reversal about one fixed constant, its time average must also be spatially constant. Subtracting a different average at each point changes the observable being tested.

In the linked U movie, 89.65% of time-varying energy lies in the fundamental, but 8.70% lies in nonzero even harmonics; the temporal average also varies with position. Both obstruct exact global complementarity. Gray–Scott does not have a U↔V symmetry at these parameters: the two diffusion constants and reaction terms differ. Nor is U+V conserved. A display-palette resemblance is consequently not evidence of a chemical color-exchange law.

An exact derived movie is possible. Define the odd part of the observed field by

Then b(x,t+½)=−b(x,t) identically. It inherits every existing space–time symmetry because this projection commutes with spatial operations and phase shifts. Therefore the saved g6 wave's odd part has an exact (R,¾,1) relation, and thresholding b at zero gives an exact two-color version away from its nodal set. This is a filtered visualization of the orbit; it is not itself asserted to solve the original PDE.

7. How to apply the theory to the 68 polar entries

“Polar” here means the time direction is preserved, as in the site's 68 forward clockwork groups. Spatial mirrors and glides may still occur. The site's stored action list contains 51 entries with nonzero phase offsets and 17 zero-offset spatial references. Its counts classify ordinary uncolored space–time geometry; adding independent color data changes the classification problem.

1. Specify the observable. Use actual two-color labels, a signed contrast field, or an explicit fixed transformation of the palette coordinate. Record the primitive period and any neutral boundaries.
2. Test the kernel first. Determine whether the pure half-period swap exists. It decides whether each spatial operation has one lift or a pair of lifts.
3. Use the presentation to propose triples. Solve the phase/color congruences on the wallpaper generators, in the appropriate quotient. This provides a finite algebraic search for specified crystallographic actions and color group; the 68 labels alone do not determine the answer.
4. Measure the actual movie. Check all phases and spatial points, separately per field channel or label. Check closures and primitive period; record RMS, maximum error, and mismatches. Search outside the nominal group for extra rotations and translations, since a saved field may have a larger symmetry group.
5. Classify only after validation. Quotient equivalent decorated actions by spatial coordinate changes, allowed time conventions, and global color relabeling. Distinguish prescribed symmetry, measured full symmetry, and approximate symmetry.

The natural next catalog is therefore a table attached to each saved movie: ordinary verified action, extra spatial symmetries, pure phase/color kernel, generator triples, and numerical residuals. This report supplies the theorem, a complete exact 442 illustration, and a reproducible audit of the motivating movie; it does not claim an exhaustive maximal-symmetry audit of all saved animations.

Is there a “magic theorem”? The closest elementary answer is the character/lift construction above. The established H/K theorem also relates the setwise and pointwise symmetries of a periodic orbit; for finite symmetry groups their quotient is cyclic. Its realization theorem includes representation and connectivity conditions and concerns an appropriate equivariant dynamical system. It does not guarantee a periodic solution of a particular Gray–Scott PDE, or an arbitrary color operation on its concentrations.