Crystallography & Symmetry
patterns derived from their groups
Series

Friezes — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · classification
  2. Why seven — all 16 candidates. Every subset of the 4 extras available on a strip, closed under composition and named from the operations that come out. 16 candidates give 7 distinct groups: 9 of them generate operations they were not given and land on a group already listed.

    Why sixteen become seven

    Four extra operations give sixteen combinations and seven groups. The nine that vanish are not cases anybody forgot — each one comes back from the closure holding something it was never given, and one of them changes the lattice underneath it.

    part 2 · classification
  3. 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.

    part 3 · classification

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