Crystallography & Symmetry
patterns derived from their groups
Concept

Strip — where it appears

A region of the plane infinite in one direction and bounded across it, which is what a frieze group acts on. Taking one row of a wallpaper pattern and keeping the symmetries that leave it in place gives a frieze, and which of the seven is a fact the plane group's symbol does not state.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The seven frieze groups. Every way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.

Seven friezes

The same classification argument on a strip instead of a plane, where it is short enough to check by hand. Seven ways to repeat a motif along a line, with names like hop, step and sidle.

classification · Friezes The friezes inside the seventeen. 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.

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.

classification · Friezes

Named alongside it

The objects these essays reach for when they reach for this one.

ClassificationFrieze groupForcingGlide reflectionp31mp3m1StabiliserSubgroupWallpaper group

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