Two-Color Patterns

Grünbaum and Shephard count 46 two-colour symmetry groups of the plane and 88 two-colour pattern types. Both numbers are correct, and the second is not a finer classification of groups: a pattern type is a colour group together with one further datum — where the motif sits. This page defines that datum, gives it a one-glance notation, recomputes the 88 from scratch, and lists them all with plates, presentations and crops of the two books.

1. Patterns, colourings, groups

A (discrete, periodic, monomotif) pattern 𝒫 in the sense of Tilings and Patterns, Chapter 5, is a family of pairwise disjoint congruent copies M_i of a compact connected motif M, on which the symmetry group Γ = S(𝒫) — a wallpaper group — acts transitively. The copies are therefore the coset space Γ/S, where S = S(𝒫, M) is the stabiliser of one copy, the induced group. Because copies are disjoint and connected, S is the full stabiliser of a point of the plane — the site symmetry of the point where the motif "sits" (a connected set invariant under a rotation about p separates p from infinity, so two of its images under the stabiliser of p would meet; hence that whole stabiliser fixes the copy): trivial in general position, d1 on a mirror, c_n at an n-fold gyration point, d_n at a kaleidoscopic corner. Grünbaum and Shephard call the pattern primitive when S is trivial.

A k-colouring is a map χ from the copies to k colours; a colour symmetry is a pair (s, θ) of a symmetry s ∈ Γ and a permutation θ of the colours with χ(sM_i) = θ(χ(M_i)) for every copy; the colouring is perfect when every s ∈ Γ admits such a θ (which is then unique), and k-chromatic when the θ's act transitively on the colours. For a perfect 2-chromatic two-colouring the colour-preserving symmetries form an index-2 subgroup H ≤ Γ, and the colour group is the pair Γ ⊃ H — the object of catalogue D: 46 of them for k = 2, denoted p4m[2]₃ by Grünbaum–Shephard and ∗442/2∗22 by Conway–Burgiel–Goodman-Strauss (full group / colour-preserving subgroup; for two colours the kernel K equals H).

2. Why the group is not enough

Take Grünbaum and Shephard's own witness, Figure 8.3.2 (below): two colourings of the same pattern of type PP15 — pmm with the motif seated on a mirror — with the same colour group pmm[2]₁ = ∗2222/∗2222 and the same induced group d1, which are nevertheless of different pattern types (PP15[2]₁ and PP15[2]₁*). Looking at the two pictures, the natural first reaction of a group theorist — it was Chaim Goodman-Strauss's, in conversation, August 2026 — is: that is the same group, just a different choice of generators; the mirrors are oriented differently; if an automorphism preserves the colours we would call the two colourings the same. That is exactly right, and it is exactly the point.

Both pictures have the identical group Γ = ∗2222 and, after choosing coordinates, the identical colour-preserving subgroup H — three of the four mirror generators P, Q, R, S preserve the colours and one, say S, interchanges them; the short colour signature is ∗¹2¹2¹2²2 in both. Nothing about the pair (Γ, H) distinguishes them. What distinguishes them is a third subgroup: the stabiliser S(M) of a copy of the motif, ⟨P⟩ in one picture (the seat mirror is adjacent to the swap-mirror S, hence perpendicular to it) and ⟨Q⟩ in the other (opposite to S, hence parallel). The automorphism of ∗2222 that turns the rectangle by a quarter turn (every automorphism of a wallpaper group is affine, §3) carries ⟨P⟩ to ⟨Q⟩ — but it carries H to a different index-2 subgroup: the swap-mirror moves from S to P. And no automorphism does better: any automorphism taking ⟨P⟩ to ⟨Q⟩ interchanges the two families of parallel mirrors, so it moves the colour-interchanging mirror S into the family of P and R and cannot preserve H. So the two pictures are related by an automorphism of Γ that is not an automorphism of the coloured group (Γ, H): the same group up to automorphism, the same colouring up to automorphism, but not the same pair (colouring, seat) up to any automorphism. Group theory does distinguish the two pictures; one only has to ask it about the right object.

3. The invariant

The right object is a chain of three groups. Fix a perfectly two-coloured pattern and a copy M of its motif; write

S(M) ≤ H ≤ Γ

for the stabiliser of the copy, the colour-preserving subgroup, and the full symmetry group. The inclusion S(M) ≤ H is forced: a symmetry that fixes a copy fixes its colour. Every isomorphism between wallpaper groups is conjugation by an affine map (Bieberbach), so:

This is the bijection form of the Definition rewritten: the coloured copies with the Γ-action are the Γ-set Γ/S(M) → Γ/H (copies over colours), a "compatible colour-preserving bijection" is an isomorphism of such Γ-sets, and isomorphism classes of such data are precisely conjugacy classes of chains. The three clauses — colour group, induced group, motif-transitive subgroups — are what the chain determines: (Γ, H); S(M) with its action on Γ/H; and the family of subgroups Γ′ with Γ′S(M) = Γ.

The invariant as a picture: a marked signature

Because S(M) is a point stabiliser, it can be read off Conway's orbifold of Γ: S(M) is the local group of the point where the motif sits — the interior (trivial), a mirror edge (d1), a corner (d_n), a cone point (c_n). Conway, Burgiel and Goodman-Strauss's annotated signature names every one of these strata: in ∗^P6^Q3^R2 the letters P, Q, R are the three mirror edges of the kaleidoscopic triangle, the digits 6, 3, 2 its corners (each between the two letters around it), and in ^α2^β2∗^P the letters α, β are gyration points. Their colour signature ∗²6¹3¹2 records the colour group by writing above each generator the order of its colour permutation (1 preserves, 2 interchanges). Hence:

In crystallographic language the strata refine the Wyckoff positions of Γ: a Wyckoff position is a Γ-class of site-symmetry groups, a stratum a Γ-class of edges, corners or centres, and the two differ only where one mirror line carries several edge types (p3m1: P, Q, R; p6m class b: Q, R) — 75 strata against the 72 Wyckoff positions of the 17 plane groups. Up to the affine normaliser: 52 strata classes, 51 Wyckoff sets, and the latter are Grünbaum and Shephard's PP1–PP51 with PP48A and PP48B as one set. Up to the same PP48A/B split, a two-colour pattern type is a Wyckoff set of the two-colour group whose site symmetry is entirely colour-preserving.

4. Ambient group, objects, equivalence: where the 88 fit

Every catalogue on this site follows one recipe: choose an ambient group of allowed symmetries and an equivalence relation, then count the discrete subgroups with a full-rank translation lattice up to that relation. Colour groups are subgroups of E(2) × S₂ (spatial isometry paired with a colour permutation) up to conjugacy by Aff(2) × S₂. Pattern types are not subgroups of any ambient group up to a coarser or finer conjugacy: PP15[2]₁ and PP15[2]₁* can be realised with the literally identical subgroup of E(2) × S₂. The objects have to carry the seat. Two ways to say it — the second is the chain, and gives 87; the first reproduces Grünbaum and Shephard's 88 for two colours:

So the honest answer to "is it a different way of grouping groups?" is: almost. It is a way of grouping pairs of groups — a group and a finite subgroup, or a group and one of its orbits — under the same affine equivalence as before. That is why the counts jump (46 → 88) without any group being subdivided: the group ∗2222/∗2222 is one class, and it carries four seats (general position, on P, on Q, at the corner PQ), each a type. The uniform notation on this page is therefore colour type · marked colour signature: ∗2222/∗2222 with ∗¹2¹2¹2²2 marked at P. (Chapter 5's varieties are a positional refinement of the same kind, and they are what separates 88 from 87.)

5. The count, recomputed

The enumeration (enumerate/enumerate_patterns.py, exact rational arithmetic, no dependencies) starts from the models of the 17 wallpaper groups used by catalogue D, defines Conway–Burgiel–Goodman-Strauss's generators for each group in those coordinates and verifies their presentations exactly, computes the strata of each orbifold from the annotated signature (edges as mirror segments between consecutive corners, corners as fixed points of products of adjacent mirrors, gyration centres), enumerates the index-2 subgroups H, keeps the pairs (stratum, H) whose local group lies in H, and takes orbits under the affine normaliser. Result: 52 strata classes (51 Wyckoff sets), and for two colours 88 marked types over the 46 colour groups — 46 primitive (general position: one per colour group, exactly Figure 8.2.2) and 42 non-primitive (exactly the 42 panels of Figure 8.3.5) — and 87 chains, PP48A and PP48B merging. The G&S labels are assigned by the stabiliser type read off the plates (PP12 = pmg with c2, PP13 = pmg with d1, PP24 = p31m with c3, PP38/PP39 = p4m with d1 on an axis/diagonal mirror, PP47/48A/48B = p6m with d1 on P/Q/R, and PP15[2]₁ versus PP15[2]₁* by the seat mirror being perpendicular or parallel to the colour-interchanging mirrors) and cross-checked against the list of catalogue E plate by plate.

6. Reading an entry

The notation. Every pane carries the three groups of its type as G / H / S(M). The first two are Conway, Burgiel and Goodman-Strauss's own letters: The Symmetries of Things §12 (p. 155) introduces, beside the full group G and the kernel K of symmetries fixing every colour, "a third group H, consisting of the symmetries that fix any one chosen color… called the stabilizer of that color", and writes the colour type as G^p/H/K, simplified to G^p/K when H = K and to G^p//K when they differ. With two colours the stabiliser has index 2, hence is normal and equals the kernel, so the single-slash form always applies: our G/H is the G/K of Table 11.1 — the value printed in the “Chaim colour type” row of the same pane, checked against that table for all 88 — and the book itself writes "color type G/H" for twofold colourings in §11 (pp. 144, 149). The third term is not theirs: S(M) is the stabiliser of one copy of the motif, Grünbaum and Shephard's induced group, written here by its generators — ⟨P⟩ ≅ d1 on a mirror, ⟨P, Q⟩ ≅ d2 at a corner, ⟨α⟩ ≅ c3 at a gyration point, 1 in general position. It is the only one of the three that varies between the types over one colour group, and it is what takes 46 to 88.

Two things the string does not do on its own. Its middle term is an isomorphism type, and a type can occur twice over one group: pm has two different index-2 subgroups both of type pm, so ∗∗/∗∗ names two colour groups — the one whose colour-reversing element is a mirror and the one where it is the translation. That is the single exception the book itself records ("There is only one case in which the symbol G/K does not completely specify the color type", p. 139), and its Table 11.1 tags the rows ∗∗/∗∗(1) and ∗∗/∗∗(2); the panes carry that tag, and with it the printed string separates all 88 types. Second, S(M) as an abstract type would be far too coarse — ⟨P⟩ and ⟨Q⟩ are both d1 but sit differently against the colour-interchanging symmetries, which is the whole content of §2 — so the seat is printed by its generators, and the marked colour signature below it remains the primary label. Even the generators do not separate everything at the group level: ⟨Q⟩ and ⟨R⟩ in p6m are conjugate in Γ, so PP48A and PP48B print different strings but have identical chains, which is exactly the 88-against-87 gap of the aside above.

One dictionary note, since two notational systems meet here. Grünbaum and Shephard have no letters G, H, K: they write S(𝒫) for the symmetry group of the pattern — our G — and S(𝒫̂) for the colour-preserving group of the coloured pattern — our H at two colours — and they name the motif stabiliser the induced group without giving it a symbol; S(M) is this page's abbreviation for it. And in the colour-symmetry literature (De Las Peñas, Felix and others) G ⊇ H ⊇ K is standard with a different H: there G is the symmetry group of the uncoloured figure and H the subgroup of symmetries that permute the colours at all, the stabiliser of one colour being called J. Here every symmetry permutes the colours — the colourings are perfect by construction — so that H would be all of G. Ours is the book's H: the stabiliser of a colour.

The catalogue is grouped by wallpaper group, then by colour group (a band with the colour type and signature, linking to catalogue D and to the Table 11.1 row in The Symmetries of Things), then by seat. Each pane shows: the plate — one motif copy per coset of S(M), drawn with the symmetry of S(M) (an asymmetric R-diamond in general position, an arrowhead on a mirror, a parallelogram at a half-turn, a rectangle at a d2 corner, pinwheels and regular polygons for higher orders), coloured by H, with the generators of Γ overlaid (solid: preserves colours; dashed: interchanges them) and the seat ringed; the data table with links to the Grünbaum–Shephard plate crop (↗) and to the Chaim row (↗); and the Presentation, Chaim's generators with their colour permutations, augmented with the pattern-type invariant: which generators fix the motif (●), the chain S(M) ≤ H ≤ Γ, and the sibling types over the same colour group. Notation: ∗ is typeset for the book's kaleidoscope star; subscripts follow Grünbaum–Shephard's Table 8.2.2 (pmm[2]₄) and the Shubnikov symbol is given for orientation.

B. Grünbaum, G. C. Shephard, Tilings and Patterns, Freeman 1987 — Chapter 5 (patterns, PP1–PP51), Chapter 8 (§8.2 colour groups, Figure 8.2.2; §8.3 colour pattern types, Figures 8.3.2, 8.3.5, statements 8.3.1–2). J. H. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries of Things, A K Peters 2008 — Chapter 10 (presentations from signatures), Chapter 11 (twofold colourations, Table 11.1). International Tables for Crystallography A, §8.3 (Wyckoff positions and Wyckoff sets). The clockwork/colouring correspondence and the presentations of the 46 colour groups follow the companion Catalog of colorings.