Shubnikov

Each of the 990 classes gets its own complex field, a sum of 12 to 72 travelling plane waves

Ψ(x, t) = Σ_p b_p exp(2πi(p·x + ν_pt)),

with x in lattice coordinates and t in film periods. The wavevectors p lie in a ring around one wavelength — about two waves to a lattice edge — as a Turing instability selects; the frequencies ν_p lie in ν₀ + {−1, 0, 1} cycles per period, locked to the film's clock, as a Hopf bifurcation does. Plane waves are the normal modes of every linear reaction–diffusion or wave equation, so these pictures speak that alphabet near a Turing–Hopf onset. They are not simulations of any one named equation — nothing here solves Gray–Scott or the Brusselator. For waves integrated from a chemistry, see the Monochrome atlas and the colour pages.

Colour is phase. A point takes colour k when the phase of Ψ there falls in the k-th n-th of the circle. The coefficients are set so that Ψ(g·z) = e^2πiσ(g)/n Ψ(z) for every symmetry g of the film, which turns the phase by exactly σ(g) sectors: g carries colour k to k + σ(g) exactly, not to within a pixel. One period multiplies Ψ by e^2πic_t/n, so the coloured film repeats after exactly m periods.

Two views. In greys the n colours are n evenly spaced lightnesses, dark at colour 0 and light at colour n − 1. The one colour view paints colour 0 near-black and the rest near-white, so the black set has exactly the symmetry of ker σ, the index-n subgroup that keeps the colours.

The audit. A field obeying the law can still carry symmetry nobody asked for. So the builder works in exact rational arithmetic — each orbit of waves takes a distinct amplitude and a phase of k/7919 of a turn — and solves, over every isometry of the lattice, with and without time reversal, on Ψ and on its conjugate, for all constant-phase symmetries. A picture is kept only if those are exactly the elements of G, with shifts exactly σ. 798 of the 990 passed on the narrowest ring; 192 needed a wider one. In the worked case, g128 over p4m with all three mirrors swapping colours, a narrow ring leaves a spurious half-cell antitranslation, and the ring widens until the next shell of waves breaks it.

Set n = 2. A two-colouring of G is a homomorphism onto ℤ₂, which is the same thing as a subgroup of index 2 — the symmetries that keep the colours. Every index-2 subgroup is normal, so at two colours "cyclic" costs nothing: this slice is the whole two-colour theory, Shubnikov's antisymmetry, black and white, or, in physics, time reversal.

Heesch (1930) counted the 122 "four-dimensional" point groups of ordinary space as abstract mathematics; Shubnikov's Symmetry and Antisymmetry of Finite Figures (1951) has 32 ordinary + 32 grey + 58 black-and-white point groups; Zamorzaev (1953) and, independently, Belov, Neronova and Smirnova (1955) reached the space groups, in a paper titled "1651 Shubnikov groups": 1651 = 230 ordinary + 230 grey + 1191 black-and-white.

That gives a check. The 68 forward film groups are exactly the 68 polar space groups, read with the polar axis as time, and over those 68 the magnetic-group tables list 304 black-and-white groups: 129 of type III, where no translation swaps the colours, and 175 of type IV, where one does. This census gives 309 or 307, depending on which changes of coordinates are allowed.

The five splits. In P1, Pm and Cc the polar direction is not unique: a crystal has more than one axis it could call time and need not choose, while a film must. "The colours swap every period" and "the colours swap from one stripe of the pattern to the next" are two different films, and a crystal may turn one into the other by relabelling axes. So P1 gains a class here, Pm three and Cc one. Nothing is lost by it: every one of the 309 lands on one of the 304, every one of the 304 is hit, and 65 of the 68 groups agree exactly.

The two merges. E_fwd, the version shipped here, also allows a plain mirror of spacetime, so a colouring and its mirror image count as one. That merges two enantiomorphic pairs of colourings the magnetic tables keep apart: P_c4₁ with P_c4₃ over P4₂, and P_I4₁ with P_I4₃ over I4₁ — the whole of the gap between 309 and 307.

The 237 two-colourings the clock cannot see are plain black-and-white patterns of the plane. Between them they realise all 46 two-colour wallpaper types. Adding the clock-invisible colourings with three, four and six colours, every cyclic plane colour group with two, three, four or six colours appears, 80 in all. The other 683 classes here use three, four or six colours, where Shubnikov's tables do not reach.

More than two colours arrived quickly. Belov and Tarkhova, "Groups of coloured symmetry" (1956), let a symmetry cycle through p colours rather than swap two — the cyclic case, which is the case on this page. Then Indenbom (1959) and Niggli (1959) gave it its proper setting: colour groups come from the one-dimensional representations of the group. A real one, with values ±1, is a black-and-white group; a complex one is polychromatic. A map σ: G → ℤ_n is exactly a one-dimensional representation with values in the n-th roots of unity, which is also why the picture here can be a single complex field with the colour read off its phase.

Koptsik (1966) published the atlas of all 1651, and Shubnikov and Koptsik's Symmetry in Science and Art (Nauka 1972; Plenum 1974) carried the subject to a general readership. For three-dimensional space groups beyond two colours the papers are Harker, Acta Cryst. A37, 286 (1981), for three colours, and Sivardière A40, 573 (1984), Roth A41, 484 (1985) and Sivardière A44, 735 (1988) for four and six.

Their counts have not been read, so nothing at n = 3, 4 or 6 on this page is checked against the literature. The 683 classes beyond two colours stand on this site's own arithmetic; the two-colour comparison above is the only external anchor.

One card is one class, not one film. A film appears once for every colouring it carries, so g10 appears 35 times and g247 four.

the film

its number in the 275 (g128), its wallpaper group (p4m), and the 3D polar space group it is when time is read as the polar axis (P4mm). The 275 are in the catalogue; the 68 forward ones are laid out by wallpaper group on the correspondence pages.

n

how many colours: 2, 3, 4 or 6.

c_t

σ of one film period — how far the colours advance while the film runs once through. c_t = 0 means the clock changes nothing.

m

n/gcd(c_t, n): how many periods of the film the coloured film takes to repeat. The card loops m periods.

the greys

colour 0 darkest, colour n − 1 lightest, evenly spaced between. A boundary is where the phase crosses from one n-th of the circle to the next.

one colour

the second view: colour 0 black, the rest white. What you are then looking at is ker σ, the index-n subgroup that keeps colour 0 where it is.

Two cards of the same class, drawn from different seeds, would look different and mean the same thing.

Two colourings are the same when a change of coordinates carries one to the other and the colours can be renamed to match. Which changes of coordinates are allowed is a real choice, and it changes the total.

• E_fwd — any affine map of spacetime that keeps the direction of time. Mirrors of space are allowed, and so are shears that tilt space against time, which is only a change of moving frame. 990 classes: 307 at n = 2, 89 at 3, 241 at 4, 353 at 6.
• E_p3 — only maps of determinant +1 in three dimensions. A bare mirror of space is dropped; a mirror taken together with a reversal of time is admitted in its place, since the two minus signs cancel. It is therefore not a narrowing of E_fwd: on the eight screw films whose mirror image is a different film it allows twice as much. 1006 classes: 309, 93, 243, 361.

Either way the equivalence acts inside one film group, on that group's colourings: a colouring and its mirror image may be one class. Which film is which is never in question — P4₁ and P4₃ are two of the 68 under both counts, and so are P3₁/P3₂, P6₁/P6₅ and P6₂/P6₄.

The two disagree in exactly 14 of the 272 (film, n) cells, all in the three chiral families p4, p3 and p6. What is chiral in those cells is the colouring, not the film: the film group has no mirror of its own, so σ and its mirror image can be two different rules, and E_fwd, which allows the bare mirror, merges them where E_p3 does not. The films that really do come in a left hand and a right hand are exactly the ones where nothing differs at all.

• p4, six cells: P4 at n = 4 (9 against 10), P4₂ at n = 2 and 6 (4 against 5, twice), I4 at n = 4 (5 against 6), I4₁ at n = 2 and 6 (2 against 3, twice).
• p3, four cells: P3 at n = 3 and 6 (5 against 6, twice), R3 at n = 3 and 6 (3 against 4, twice).
• p6, four cells: P6 at n = 3 (3 against 4) and at n = 6 (9 against 12), P6₃ at n = 3 and 6 (3 against 4, twice).

The week-38 review recommended E_p3, on the grounds that crystallography keeps enantiomorphs apart and a film should too. Representatives were built and audited for E_fwd only, so this page shows 990 and says which equivalence it is showing. The difference is small, listed above in full, and not hidden.

Every cyclic colouring of every forward film group, counted up to E_fwd. The last column is the colourings a clock cannot see: one period of the film leaves every colour where it was.