Crystallography & Symmetry
patterns derived from their groups
Concept

Rotation — where it appears

An operation turning the plane or space about a fixed point or line, whose order is how many of them return everything to where it started. In a lattice basis it is an integer matrix, which is why only the orders one, two, three, four and six survive.

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

A glide. The motif in the first colour, its images under a single glide in the second, and the symmetry element marked where the operation itself says it lies.

The four motions of the plane

Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.

operations · What symmetry is The 4₁ screw axis. 1 of the eleven screw axes a lattice permits, each drawn as the helix it is: 4₁. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

Turning and climbing at once

A rotation has a fixed point and a screw has none. That sounds like a small difference and it is the reason a space group is not a point group with extra letters, the reason two hundred and thirty is not seventy-three, and the reason a helix can be a crystal.

space-groups · Screws and glides

Named alongside it

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

ChiralityFixed pointGlide reflectionHelixIntrinsic translationLattice translationOrderReflectionScrew axisTranslation

All concepts