Patterns

A perfectly k-coloured pattern type is a chain of three groups S(M) ≤ H ≤ Γ, up to affine equivalence: Γ the wallpaper group of the pattern, H the stabiliser of one colour (the pair Γ > H, of index k, is the colour group), and S(M) the site symmetry of the seat where the motif sits. Every pane below is one such chain: the plate is computed from the exact data in data/patterns.json (by enumerate/enumerate_patterns_k.py), the red ring marks the seat, and the red index in the Chaim short form marks the symbol whose symmetry fixes the marked motif — and therefore preserves its colour.

Anchors: the colour-group counts are Grünbaum & Shephard’s Table 8.2.1 (= Wieting); the 88 two-colour types are Figs. 8.2.2/8.3.5 and the three-colour types Figs. 8.2.3/8.3.6 of Tilings and Patterns; colour groups follow Table 11.1/12.1 of Conway, Burgiel & Goodman-Strauss, The Symmetries of Things. Book crops are reproduced for scholarly commentary. Beyond the book: the exact enumeration gives 64 three-colour types where the book’s figures show 59 — five types (all with mirror-edge seats in non-regular colourings, marked on their panes) are missing from Figs. 8.2.3/8.3.6; the five panes are pmm, cmm, p3m1, p6m, p6m′. No published tables exist beyond three colours, so the labels for k ≥ 4 (and the group symbols hm[k]ₙ) are systematic extensions in catalogue-D order. The earlier two-colour study remains as the detailed treatment of k = 2.