Crystallography & Symmetry
patterns derived from their groups
Concept

Rotation order — where it appears

How many times an operation must be repeated to return every point to its start. A lattice permits only the orders one, two, three, four and six, because the trace of an integer matrix is a whole number and 2cos(2π/n) rarely is.

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

The five rotations a lattice will carry. One motif and every rotation a plane lattice permits: orders 1, 2, 3, 4, 6, and nothing else up to 12. Each panel turns the motif by its own operation as many times as the order allows, on the lattice that operation requires — oblique for the identity and the half turn, hexagonal for the third and the sixth of a turn, square for the quarter. The trace printed under each is the sum of the diagonal of the operation's matrix written in the lattice's own basis, and it is a whole number in every panel, which is the entire content of the crystallographic restriction. The list of orders is produced twice, once from that trace condition and once from the degree of a cyclotomic polynomial, and the figure refuses to draw if the two disagree.

The crystallographic restriction

A repeating pattern may have rotations of order two, three, four or six, and nothing else whatever. The proof is one line of arithmetic, and everything finite in the subject descends from it.

restriction · Restriction Assuming a 5-fold rotation. The shortest lattice vector, its rotated copies, and the combination of them that is itself a lattice vector. Where that combination comes out shorter than the vector assumed shortest, the assumed rotation cannot exist.

Why five-fold is impossible

A second proof, geometric rather than algebraic: assume a five-fold centre, and out of it construct a lattice vector shorter than the shortest one there is.

restriction · Restriction The icosahedral group, counted. The sixty rotations of an icosahedron, found by trying every map that sends one adjacent pair of vertices to another and keeping those that carry the whole vertex set onto itself. They fall into 6 axes of order 5, 10 axes of order 3, 15 axes of order 2 — and the fivefold axes are the reason this group cannot be the point group of any crystal, since no three-dimensional lattice admits a rotation of order five. Quasicrystals have it anyway, which is what made 1982 an argument rather than a measurement.

Icosahedral symmetry

Sixty rotations, six fivefold axes, and no lattice in three dimensions that can hold any of them. It is the point group a crystal is forbidden, and the one the first quasicrystal turned out to have.

aperiodic · Quasicrystals

Named alongside it

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

Crystallographic restrictionQuasicrystalDiscretenessHigher-dimensional latticeHolohedryIcosahedral symmetryIndexingInteger matrixPoint groupProof by contradictionShortest vectorTrace

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