Crystallography & Symmetry
patterns derived from their groups
Concept

Cayley graph — where it appears

The graph whose vertices are a group's elements and whose edges join each element to the elements one generator away. For a wallpaper group it is a crystal net, so counting words and counting neighbours of an atom turn out to be one computation.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The ball of radius 5 in p6. Every element of p6 reachable in at most 5 multiplications by a generator or its inverse, plotted at its translation part — so each dot is a lattice position and its size says how few steps reach it. The picture is the word metric's unit ball scaled up, and its shape is what fixes the growth: a diamond where the group supplies two short translations, and a hexagon where it supplies three. Every dot here required the word problem to be solved, because the search has to know when two products are the same element.

Telling two words apart

There are finitely presented groups in which no algorithm can decide whether two products of the generators are the same element. The seventeen are not among them, and the procedure that settles it is short enough to state in a sentence — which then makes it possible to measure how fast each group grows.

operations · Presentations 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 One curve falls and the other does not. The boundary's share of a ball, against the radius, for a plane group and for the free group on two generators. The plane group's falls like one over the radius and goes to zero; the free group's rises to two thirds and stays. A group with no sequence of regions whose boundary becomes negligible has no shape-independent average, and that is not a difficulty in the analysis — it is a property of the group.

The boundary a growing region forgets

Quoting a density assumes the region it was averaged over does not matter, and that assumption is a property of the group of translations rather than of the crystal. A ball in a plane group grows like R² and its boundary like R, so the edge becomes negligible — and where that fails, the average genuinely moves. The free group on two generators keeps two thirds of itself on the boundary forever, and a slab seven layers deep is wrong by exactly one seventh however wide it is made.

operations · Presentations

Named alongside it

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

Generating setAbelianisationAmenabilityCrystal netDecidabilityFundamental domainGroup closureGroup invariantGroup presentationGrowthGrowth functionGrowth rate

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