# Rotation centres in the Gray–Scott overlays > **Scope.** This document describes the original 442 (p4) Gray–Scott catalog and its solver. The site now covers all 17 wallpaper groups and two further equations; see [research/wallpaper/README.md](research/wallpaper/README.md) and [research/equations/README.md](research/equations/README.md). Counts below refer to the 442 catalog at the time it was written. The overlay shows rotation centres, including the equivalent centres generated by the selected group. The original display repeated only the three named representatives α, β and γ by integer cell translations. Those representatives generate the abstract group, but their displayed positions were an incomplete map of its rotation centres. ## Required centres of the selected group Counts refer to one simulation coordinate cell, with opposite boundaries identified. | Group | α | β | γ | Total | | --- | ---: | ---: | ---: | ---: | | 442: g94–g97 | 1 | 1 | 2 | 4 | | 442: g98–g99 | 2 | 2 | 4 | 8 | | 632: g243–g248 | 1 | 2 | 3 | 6 | For a named centre c and a group operation h(x)=Ax+b, the conjugate centre is Ac+b modulo integer translations. Its time offset is unchanged: the time character is abelian and all operations preserve time direction. The marker’s glyph is rotated with h, so triangles and half-turn lenses also form the appropriate wallpaper pattern. At a centre c, the affine rotation is x ↦ Mx+(I−M)c. Selecting a marker uses this operation and its own prescribed phase, including for newly added centres. The selected centre is preserved in shared URLs. The generator checkbox remains off by default. ## Additional centres of individual solutions Some numerical solutions have smaller spatial repeat cells than the selected group requires. An offline audit checks same-time translations against **both concentrations at every saved phase**. FFT autocorrelation screens possible translations; direct comparisons of the complete saved Float32 movies decide which pass. The limits are maximum absolute error 10⁻⁷ and RMS error at most 10⁻⁶ of each channel’s space–time standard deviation. The evidence is tied to the field’s SHA-256 hash. Of 207 current records, 69 have additional spatial periods compatible with their sampling mesh: 38 of 143 square-lattice records and 31 of 64 triangular-lattice records. All accepted translation sets are closed under addition. Their largest measured pointwise error is 2.98×10⁻⁸; 601 of the 640 accepted translations, including identity translations, are bit-for-bit exact. An independent check also evaluated all 1,500 distinct derived affine rotation/phase operations (representing 3,448 centres) on every saved frame and grid node of both concentrations. The largest absolute error was 2.98×10⁻⁸; the largest RMS error was 2.45×10⁻⁹. The cited g248 record at F=0.00404, k=0.02, spatial shell 3 has exact additional periods (1/3,2/3) and (2/3,1/3) in triangular lattice coordinates. Its original computational cell contains **3 sixfold, 6 threefold and 9 half-turn centres**. The denser overlay therefore contains 18 centres per simulation cell. Extra centres are found by solving (I−M)c=v+d+n for certified same-time translations d and integer vectors n. At coincident centres, the highest rotation order is shown. This is necessary because a centre originally marked as threefold can be sixfold for a more symmetric solution. Merely translating the old three markers would not find all the new centres. White markers belong to the selected group’s complete centre map. Mint markers denote additional centres implied by the checked periods of the current solution. These are numerical properties of the saved fields, not new imposed equations or a complete classification of continuum symmetries. Only periods that preserve the actual spatial interpolation mesh enter the solid overlay. Browsing uses the saved audit; it does not rerun the search or integrate the PDE. ## Apparent repeats that fail the strict check Both viewers also offer **approximate-centre inspection** when an offline audit identifies a smaller near-periodic lattice. This inspection is available with the generator overlay, which remains off by default. When generators are enabled, approximate centres are included unless the user unchecks them. Dashed amber markers are explicitly approximate; they are never added to the strict symmetry certificate or used to admit a solution. The concentration samples, playback interpolation, and prescribed group operations remain unchanged. Two reported g247 examples illustrate the distinction: - **Q12, N=96, L=1024:** the saved field has 12 exact same-time translations modulo the simulation lattice. Its strict overlay contains 12 sixfold, 24 threefold and 36 half-turn centres: 72 in total per simulation cell. - **Q31, N=66, L=1024:** the strict translation subgroup contains only the identity, hence six strict centres per simulation cell. The visually repeated motifs suggest a smaller lattice of index 31, but its translations are not mesh-compatible and the saved movie does not have those exact periods. The optional inspection shows its 186 proposed centres, with the 180 additional ones dashed. The worst translation mismatch in the actual interpolated movie is about **3.53%** of a concentration's full range. The offline audit examines all 207 fields: 16 square-lattice and 14 triangular-lattice fields have additional approximate repeats. Proposals come from dominant spatial Fourier modes, not pattern names or screenshots. Every translation in a proposed finite subgroup must be measured; the subgroup must contain the strict translations and close under addition and the appropriate 90° or 60° rotation. Both concentrations must stay below 5% maximum error and 2% sample RMS, relative to each concentration's range over the full movie. These are thresholds for displaying an approximate relation, not for calling it a symmetry. The maximum is evaluated at all vertices of the common refinement of the original and translated playback meshes, at every saved phase. The difference is affine on each triangular piece or bilinear on each rectangular piece; both attain their absolute maxima at vertices. These checks bound all spatial points; linear temporal interpolation cannot increase the maximum. Reported RMS is a finite-sample RMS at those vertices, not an area integral. Adding the measured canonical rotation residual to a translation's maximum bounds the corresponding derived rotation with its assigned time offset. Selecting a dashed marker displays its actual comparison error and explicitly states that it is not verified at the strict tolerance. Where an approximate higher-order centre coincides with a strict lower-order one, a solid outer ring and the selected-centre description preserve that distinction. Shared links include `approx=1` or `approx=0` to preserve this inspection preference. ## Pattern cells and shared framing The default selector now displays **1 cell**, **2 × 2 cells**, or **3 × 3 cells**. These are actual blocks of the pattern's primitive translation cell, not multiples of the original simulation size L. Every block is outlined, and the field is clipped to its outer square or rhombus so partial cells outside the selected block cannot be mistaken for extra cells. For a same-time translation subgroup of index d, a shortest primitive cell has side length L/√d. The viewer derives two primitive translation vectors from the checked subgroup. It applies one camera transformation to the saved field, CPU/GPU playback, comparison panels, cell boundaries, generator positions, and glyph orientations. Choosing a denser pattern therefore zooms in correspondingly; it no longer shrinks many motifs into a nominal one-cell view. Markers at boundaries are shared with neighbouring cells when counting centres per cell. For 442 the outlined cell is square. For 632 it is a 120° rhombus; its geometry is preserved without stretching it into a square. The camera rotates oblique primitive bases into the same display orientation. When a smaller cell is only approximate, that is stated in the dropdown and canvas label, with approximate generators drawn dashed. Scaling never changes the samples or promotes those operations to verified symmetries. The reported Q12 example uses a cell side L/√12 and Q31 uses an explicitly approximate cell side L/√31. Both now display one outlined cell at count 1. Version-1 links migrate their requested tile count to this corrected cell framing while retaining the selected pattern, palette, phase, and playback state. New links use version 2 and explicitly save `framing=cells` or `framing=simulation`, the count, and the approximate-overlay preference. **Simulation width** remains available as a separate view for reproducing the old physical crop: its 1, 2, and 3 options mean widths L, 2L, and 3L. It is not labelled as a pattern-cell count. ## Reproduction From the repository root: ```sh python3 docs/scott-gray/research/build-overlay-translations.py python3 docs/scott-gray/research/build-overlay-near-translations.py python3 -m unittest discover -s docs/scott-gray/research -p test_overlay_translations.py python3 -m unittest discover -s docs/scott-gray/research -p test_overlay_near_translations.py node --test docs/scott-gray/tests/rotation-centres.test.mjs docs/scott-gray/tests/overlay.test.mjs docs/scott-gray/tests/view-state.test.mjs ``` Implementation: [rotation centre construction](rotation-centres.mjs), [translation audit](research/build-overlay-translations.py), and [saved translation evidence](data/overlay-translations.json). The separate [approximate-repeat audit](research/build-overlay-near-translations.py) produces [inspection evidence](data/overlay-near-translations.json). [442 gallery](./) · [632 gallery](p6/)