WallpaperAll seventeen ways to repeat

Plane group 17 / 17 · Hexagonal · Order 12

p6m

sixth-turn · six mirror directions · glides

The theorem

There is no eighteenth way

Any pattern that repeats in two directions — Egyptian basketry, Baghdad brickwork, Alhambra zellige, a William Morris willow bough — is governed by one of exactly seventeen symmetry groups. Fedorov proved the census closed in 1891; Pólya rediscovered it in 1924, and Escher copied his diagrams by hand into a notebook. Not roughly seventeen. Exactly.

This plate stamps one figure — a single Fraunces ampersand, its weight and optical size slowly breathing — through every group in turn. Translations, half-turns, mirrors, glide reflections: each group is one legal combination, and the seventeen together exhaust everything the plane permits. Solid teal lines are mirrors; dashed lines are glide axes; brass marks are rotation centres.

The census

Seventeen groups, four lattices, one ampersand

Reading the plate

Construction marks

mirror line glide axis 2-fold centre 3-fold centre 4-fold centre 6-fold centre

Second read: the ampersand is chiral — it has a left hand. Wherever a solid or dashed teal line runs, half the stamps are flipped, printing left-handed ampersands you will never find in a font menu. The four groups with no teal lines at all (p1, p2, p3, p4, p6 — the rotation-only family) never flip a single one.