Colour symmetry · exact orders one through six
A forward-time film symmetry can be read as a wallpaper symmetry coloured by a circle of phases. Finite cyclic colourings are the sampled version of that clock—but only after the colour action, the spatial projection, and the spacetime equivalence are kept distinct.
Result
The 68 forward representatives in the 275-class audit have canonical displayed clock orders loading… for exact orders N = 1, 2, 3, 4, 5, 6. These are not Wieting's counts of all perfect N-colourings; the exact bridge passes through regular cyclic colour groups.
01
Let Γ be a wallpaper group and H the subgroup preserving one selected colour. A perfect N-colouring is a transitive coset action on Γ/H. It extends directly to one forward-time clock precisely when H is normal and Γ/H is cyclic of order N:
H ◁ Γ and Γ/H ≅ CN.
Equivalently, a surjection χ: Γ ↠ CN assigns each spatial operation a phase χ(γ)/N in ℝ/ℤ. The colour has become a time offset. Conversely, any forward-time group with fixed spatial projection and finite phase image supplies such a cyclic character. For N = 2 every index-two subgroup is normal with quotient C2, which is why the regular-cyclic row reproduces the classical count 46 exactly.
02
Forgetting every operation's time component places the 275 spacetime representatives into the 17 wallpaper projection categories. That does not say that a generic frame has the full projected wallpaper symmetry. Its guaranteed frame-preserving group is the zero-phase kernel H, often a proper subgroup of the projection Γ. Special frames may have larger stabilizers, and a particular motif may add accidental symmetries.
03
Wieting counts every transitive perfect colouring. The middle row keeps only normal kernels with cyclic quotient. The last row bins canonical forward-time representatives by the exact order displayed after the broader spacetime re-basing equivalence.
Loading the generated census…
04
Conway orbifold notation is primary in both tables. The downloadable JSON and detailed CSV retain the corresponding International short (Hermann–Mauguin) code for traceability. Every one of the 68 forward records occurs in exactly one cell of the second table.
05
Why five disappears
The four regular cyclic five-colour plane types carry C5 through translations, because planar crystallographic point groups have no order-five operation. Spacetime re-basing gives their classes canonical representatives in lower clock-order bins, so the displayed order-five bin is zero.
Why one class can split
Orbifold 333 has two regular-cyclic three-colour types. Its gyrational kernel yields the chiral pair 3₁3₁3₁ and 3₂3₂3₂; its sublattice kernel yields r33₁3₂. Two unoriented colour kernels therefore give three oriented spacetime normal forms.
06
A genuinely coloured film carries two pieces of chromatic data: an internal finite colour action and the phase clock. In the simplest abelian case this is a cocycle into CM × ℝ/ℤ. Time reversal adds the black–white involution, which acts on the clock by phase inversion. Non-cyclic perfect colourings—such as S3 colour actions—would require motifs with a non-abelian internal state rather than a single scalar clock.
The 51-row nontrivial clockwork/coloring correspondence gives the preceding, uncoloured case after omitting the 17 tautological one-colour products: every remaining canonical forward clock is paired with its regular cyclic plane colouring, zero-phase kernel, and traditional static wallpaper plate.
No count of those coloured spacetime groups is asserted here. Such a census must first specify whether equivalence is taken inside a fixed parent, under the full affine normalizer, or under the Galilean re-basing used by the 275-class spacetime catalog.
07
Download the generated JSON census, summary CSV, orbifold audit CSV, or the 68-record forward manifest.
python3 scripts/generate_color_forward_census.py
python3 scripts/generate_color_forward_census.py --check
python3 -m unittest discover -s tests -p 'test_color_forward*.py'The regular-cyclic cells were independently reconstructed with the pinned Senechal–Wieting subgroup engine: normal index-N kernels are grouped into affine-normalizer orbits, with stability checked across matrix-entry bounds one through six.
Loading data provenance…
08
T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982), Table 11; totals preserved as OEIS A307293. M. Senechal, “Color groups,” Discrete Applied Mathematics 1 (1979), 51–73. J. D. Jarratt and R. L. E. Schwarzenberger, “Coloured plane groups,” Acta Crystallographica A 36 (1980). J. H. Conway, H. Burgiel, and C. Goodman-Strauss, The Symmetries of Things (2008), Part II.
“Forward-time group” here means that every symmetry preserves the direction of time. The 17 labels in the audit are standard Conway orbifold symbols for the projected plane groups; the phase-decorated spacetime notation used elsewhere on this site is documented in the notation guide.