The friezes inside the seventeen — the frieze in wallpaper generator
Every plane group contains frieze groups: keep only the operations that map one lattice row onto itself and what is left is a group on a strip, which must be one of the seven. Across all seventeen plane groups and their principal directions, all seven frieze groups appear. The commonest is p2, in 9 of the 32 rows examined.
2 essays call
frieze-in-wallpaper. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever member
of the family the generator happens to default to rather than the one its essay argues about.
Every one of this site's 393 essays names its parameters at the
call site, which the standard pass of 2026-08-09 established and param-floor
holds.
Where it is called
Changing this generator changes every one of these figures.
The classification
The friezes inside the seventeen
Take one lattice row of a wallpaper pattern and keep only the symmetries that leave that row where it is. What survives is a frieze group — and which of the seven it turns out to be is a fact about the plane group that its symbol does not state.
The classification
A layer is not a wallpaper
A sheet repeats in two directions and lives in three, and its symmetry group is not one of the seventeen. There are eighty of them, the difference between one and another is a single sign per operation, and the arithmetic that supplies those signs is the arithmetic of a two-coloured pattern.