Catalogue E — the colour pattern types

Grünbaum & Shephard's picture-level refinement of the colour groups: the 88 periodic two-colour and 59 periodic three-colour pattern types, indexed plate by plate — and the edge of knowledge: for k ≥ 4 the number of k-colour pattern types is an open problem.

1. The definition

The coloured crystals of catalogue D classify colourings by an abstract invariant — the colour symmetry group C(𝒫̂), the group of pairs (s, θ) of a symmetry of the underlying pattern and a compatible permutation of the colours. Section 8.3 of Grünbaum and Shephard's Tilings and Patterns classifies the coloured pictures themselves. Their definition, shortened but faithful: two k-chromatic patterns 𝒫̂₁ and 𝒫̂₂ are of the same k-colour pattern type if — possibly after an appropriate renaming (permutation) of the colours in one of the patterns — they have the same colour symmetry groups, the same stabilisers of their coloured motifs (the induced colour groups), and the same set of coloured-motif-transitive subgroups of their colour symmetry groups. Equivalently, in the compatible-bijection form that runs through their Chapter 7: 𝒫̂₁ and 𝒫̂₂ are of the same type if, after renaming colours, there is a one-to-one colour-preserving mapping Ψ from the coloured copies of the motif of 𝒫̂₁ onto those of 𝒫̂₂ such that Ψ is compatible with the underlying group S_C(𝒫̂₁) and Ψ⁻¹ is compatible with S_C(𝒫̂₂).

This is strictly finer than the group-level classification of catalogue D, and none of the three clauses is redundant. The colour group cannot see how it sits on the picture; adding the induced group does not help either; Grünbaum & Shephard's Figure 8.3.2 is the canonical witness that the two together are still not enough:

Where do the types beyond the colour groups come from? From underlying patterns that are non-primitive — those whose motif has a non-trivial stabiliser. Over a primitive pattern the induced group is trivial and the colour group determines the type, so the plates of Figures 8.2.2 and 8.2.3 — one perfectly coloured primitive pattern per colour group — both define the 46 + 23 colour groups of catalogue D and exhaust the primitive pattern types. Every further type lives over a non-primitive pattern, where the induced colour group is non-trivial and one colour group can organise several inequivalent pictures: the underlying pattern PP15 of the figure above alone carries five 2-colour pattern types (PP15[2]₁, PP15[2]₁*, PP15[2]₂, PP15[2]₃, PP15[2]₄) over only four colour groups. Grünbaum & Shephard enumerate the non-primitive types by a merging procedure — move the coloured copies of the motif, keeping the colour symmetry group, until same-coloured copies merge — and record the result as two displayed statements:

2. Two colours — 88 periodic types

Every one of the 88 now has a full entry — plate, presentation, marked signature, book crops — on Two-Color Patterns; the pattern-type symbols below link there.

Rows follow the printed reading order of the plates; each row carries a stable anchor, and the entries of catalogue D deep-link into them. The pattern-type symbol is Grünbaum & Shephard's: the underlying pattern type (PPn in the numbering of their Chapter 5), the number of colours in brackets, and where needed a subscript matching the subscript of the colour group. The colour-group column links to the corresponding coloured crystal of catalogue D; the plate column gives figure and book page.

Primitive types — the colour groups drawn

One type per colour group: the Figure 8.2.2 panel that draws the group is the type's canonical picture.

Non-primitive types

The 42 types of Figure 8.3.5. The star on PP15[2]₁* is Grünbaum & Shephard's mark for a second, inequivalent type over the same underlying pattern with the same colour group as PP15[2]₁ — the two differ in how the induced group sits over the motifs, exactly the phenomenon of their Figure 8.3.1(c).

3. Three colours — 59 periodic types

The three-colour plates — first published, the book notes, in Grünbaum & Shephard [1983b] — occupy Figure 8.2.3 (the 23 primitive types) and Figure 8.3.6 (the 36 non-primitive ones). Two printing slips are corrected silently below: Table 8.2.3 prints the symbol of p2[3] as "p2[2]", and the Figure 8.2.3 panel of PP23[3]₂ is labelled p31m[3]₃ for p31m[3]₂.

Primitive types — the colour groups drawn

Non-primitive types

The 36 types of Figure 8.3.6, including the two starred types PP15[3]* and PP19[3]* — second inequivalent colourings of PP15 and PP19 with the colour groups pmm[3] and cmm[3] already used by PP15[3] and PP19[3]. Without them the count would be 34, not the 36 of the figure's caption.

4. Beyond three colours

For k ≥ 4 the primitive half of the classification continues without incident: primitive k-colour pattern types are exactly k-colour groups, and catalogue D counts and draws them — 96 four-colour, 14 five-colour and 90 six-colour groups. It is the non-primitive half that is missing. Nothing plays the role of Figures 8.3.5 and 8.3.6 for four colours, and the total number of k-colour pattern types is unknown for every k ≥ 4. Whoever repeats Grünbaum & Shephard's merging procedure over the 96 four-colour groups settles the first open case.

References

B. Grünbaum, G. C. Shephard, Tilings and Patterns, Freeman, 1987 — Chapter 8, §8.3; Figures 8.2.2, 8.2.3, 8.3.5, 8.3.6; Statements 8.3.1–8.3.2.
B. Grünbaum, G. C. Shephard [1983b in the book's bibliography] — cited at Table 8.2.3 for the 3-colour symbols; the paper in which the 3-colour pattern types were first illustrated.
B. Grünbaum, G. C. Shephard, Incidence symbols and their applications, in Relations Between Combinatorics and Other Parts of Mathematics, Proc. Sympos. Pure Math. XXXIV, AMS, 1979, 199–244 — [1979a]; the incidence-symbol machinery behind the colour-adjacency symbols of §8.6.