Crystallography & Symmetry
patterns derived from their groups
Concept

Generating function — where it appears

A power series whose coefficients are the counts being studied, so that a statement about a sequence becomes a statement about a rational function. The Molien series is one, with the degrees of a ring's generators visible in its denominator.

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

What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Every colour count at once

Eight hundred and five structures is the answer for two species on sixteen sites. For three species it is a different sum, and for four another. Averaging cycle counts instead of fixed-point counts turns the answer into a polynomial — and refining the same average says how many structures there are at each composition, which is the number anybody actually needs.

operations · Counting 4mm: how many independent invariants there are at each degree. The number of independent polynomial invariants of the plane point group 4mm, one bar per degree from 0 to 8. The heights are the coefficients of the Molien series, computed from an integer recursion on the trace and determinant of each operation. A dot above a bar marks a degree where the same number has been computed a second way, by averaging every monomial of that degree over the group and taking the rank of the result — a computation sharing no code with the first. The two agree at every degree checked. Degrees where the bar is absent hold no invariant at all: for 4mm that is every odd degree below the first invariant, and it is a statement about which functions the group refuses to leave alone.

How many invariants of each degree

A group moves the plane about, and some polynomials do not notice. How many independent ones there are at each degree is a sequence of integers, computed here by a recursion on traces and again by averaging every monomial — two routes that share no code and agree everywhere.

point-groups · Invariants

Named alongside it

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

Averaging projectorBurnsideCharacterColour symmetryCountingCycle indexEnumerationInvariant polynomialInvariant ringMolien seriesOrbitPattern inventory

All concepts