Crystallography & Symmetry
patterns derived from their groups
Concept

Translation — where it appears

A motion that slides everything by the same vector, and the operation a periodic pattern is built on. The set of them forms the lattice, and the whole classification of periodic patterns is a question about what else a group of them can contain.

Named by 3 essays across one field — 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 Two half-turns make a translation. The half-turn about (0.25, 0.25) followed by the half-turn about (0.75, 0.5) is the translation by (1, 0.5) — twice the vector between the two centres, and not the vector itself. The open lens is a third centre, and it is not the midpoint of the two drawn: it is where the half-turn about the first lands when it is composed with one repeat vector of the lattice, which is half a repeat along. That is the step that puts two-fold centres on the half lattice and gives a p2 cell four inequivalent ones. Both the translation and the forced centre are computed from the operations and compared with the construction in exact rational arithmetic.

Where the product is

Composing two symmetries lands on a third — and the third one is somewhere. Two half-turns make a translation by twice the distance between their centres, and that single fact puts the lattice into a pattern before anybody chooses one.

operations · Composition Four of the seventeen have a centre, and they are the four with no rotation. For each plane group: the order of its point group, how many of its operations are rotations, the lattice vectors every operation of the point group fixes, and the centre those vectors make. A central element must commute with every translation, which forces its linear part to be the identity — so the centre is a group of translations, and a translation is central exactly when the point group leaves it alone. A rotation leaves nothing alone but zero.

The four groups with a centre

An element that commutes with everything has to commute with every translation, and that forces its linear part to be the identity. So the centre of a plane group is a group of translations — the ones its point group leaves alone — and a rotation leaves nothing alone but zero. Four of the seventeen have a centre and thirteen have nothing at all.

operations · Composition

Named alongside it

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

Fixed pointAbelianisationChiralityCompositionConjugationGlide reflectionGroupHalf-turnLattice translationMirror lineNormal subgroupPlane group

All concepts