Crystallography & Symmetry
patterns derived from their groups
Concept

Quasi-polynomial — where it appears

A function that agrees with a polynomial on each residue class of its argument, with a different polynomial for each class. Lattice-point counts, coordination sequences and group growth are all of this kind, and the number of classes is called the period.

Named by 3 essays across 3 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 a lattice triangle: 1 inside, 6 on the edge, area 3. a lattice triangle on its lattice, with the 1 points strictly inside it in the first colour and the 6 points on its boundary in the measured colour. Pick's theorem says the area is the interior count plus half the boundary count less one, which is 1 + 6/2 − 1 = 3; the shoelace formula on the same integer coordinates gives twice the area as 6. The two agree, and both sides are integers, so the check has no tolerance in it. The theorem holds for a non-convex polygon and a polygon with no interior point alike, neither of which the usual triangle-and-square picture makes obvious.

How many points a shape holds

Draw a polygon on a lattice, count the points inside, then double the polygon and count again. The counts are not approximately a polynomial in the scale — they are one, exactly, with the area as its leading coefficient and a constant term of one for every polygon there is.

lattices · Lattice the kagome net: 4, 8, 14, 18 at the first four shells. The vertices of the kagome net at graph distance one, two, three and four from a chosen vertex, each marked with its distance. Distance here is a number of edges and nothing else — no length enters, and the shells are drawn on the barycentric placement only so that they can be seen. The counts are 4, 8, 14, 18, 22, 28, 30, 38, 38, 48, 46, 58, which is the net's coordination sequence.

Counting outwards

How many vertices lie one step from a vertex, two steps, three? The counts settle into a straight line — but for some nets only along the even distances, with a different line along the odd ones, alternating for ever. The period is measured, and it is not always one.

applied · Nets

Named alongside it

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

Crystal netBreadth-first searchCayley graphCoordination sequenceDilationEhrhart polynomialGenerating setGraph distanceGroup invariantGrowth functionInvariantLattice point count

All concepts