Crystallography & Symmetry
patterns derived from their groups
Concept

Generating set — where it appears

A set of group elements from which every element can be built as a product. It is never unique, so quantities that depend on it — word length, the growth coefficient, the shape of a ball — are properties of the pair rather than of the group.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

p1, p2, p4, p6m: every one quadratic. How many elements each group has at word length at most R, to 14 terms, against the same kind of generating set. Every curve is a quadratic in R — which is the group knowing its own dimension, since a crystallographic group of d dimensions grows like R to the d and nothing about the counting mentions the plane. The curves differ by a factor: p1 reaches 421, p2 reaches 786, p4 reaches 1464, p6m reaches 5478.

How fast a group grows

Take a wallpaper group, forget the plane, and keep only the generators and the rule for multiplying. Count the elements that can be spelled in at most R letters. The answer grows like R squared — for every one of the seventeen — and the group has told you the dimension of a plane it no longer knows about.

operations · Presentations p4: a domain of 38 cells with 7 walls. The fundamental domain of p4 on a grid of 12ths, with the walls it shares with its neighbouring copies marked. Each wall names the element that carries this copy onto the copy across it, and there are 7 distinct such elements. Those elements generate the whole group — checked by closing them up and requiring every coset and the whole translation lattice to be reached, not assumed — which is Poincaré's theorem, and it means the generators of a wallpaper group can be read off a picture. The domain is pixelated rather than a polygon, so the wall count is a property of this domain and not of the group.

Every wall names a generator

The copies of a fundamental domain tile the plane and stand in one-to-one correspondence with the elements of the group. So the elements that carry the home copy across a wall generate everything — and the generators of a wallpaper group can be read off a picture rather than looked up.

operations · Fundamental domain Thirty-two classes, eighteen groups. Every abstract group the thirty-two crystal classes realise, with the classes that realise it. 8 of the eighteen carry more than one class, and the largest collision is the four hexagonal classes that are all the dihedral group of order twelve. Nothing here is looked up: two classes are put in the same row when a search over images of a generating set finds a bijection preserving multiplication, and the search is finite because a generating set is small and the elements it may map to are the ones of the same order.

Thirty-two classes, eighteen groups

An inversion centre, a mirror and a two-fold rotation are three of the most different things a crystal can have, and they are the same group of order two. Forget the matrices and keep the multiplication table, and the thirty-two classes collapse to eighteen.

point-groups · Crystal classes

Named alongside it

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

Cayley graphAbstract groupCharacterClassificationConjugacyCrystal classCrystal netEquivalenceFundamental domainGroup closureGroup invariantGrowth function

All concepts