Crystallography & Symmetry
patterns derived from their groups
Concept

Reflection — where it appears

An operation that fixes a hyperplane and reverses the direction across it: a mirror line in the plane, a mirror plane in space. Its determinant is minus one and its fixed set has one dimension less than the space, which is what separates it from an inversion centre, whose determinant is also minus one and which fixes only a point.

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

A glide. The motif in the first colour, its images under a single glide in the second, and the symmetry element marked where the operation itself says it lies.

The four motions of the plane

Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.

operations · What symmetry is Four angles, and the integer that picks them. Two roots at angle θ have Cartan integers whose product is 4cos²θ. Both are whole numbers and the product is below four, so it is nought, one, two or three — and each value fixes the angle between the two roots, and with it the angle between the mirrors perpendicular to them. The shaded wedge is the region the pair of mirrors folds the plane onto; the smaller it is, the larger the group they generate.

Four root systems, and the same four rotations

Two mirrors meeting at an angle generate a group. Ask that the group be finite and that a certain pairing between the mirrors come out a whole number, and the angle has only four possible values — from which the rotations that survive are of order two, three, four and six. The crystallographic restriction arrives with no lattice anywhere in the argument.

restriction · Restriction 12 of the thirty-two classes have a free invariant ring. Every crystal class with its order, the number of its operations that are reflections, whether its ring of invariant polynomials is free, and the degrees of the generators when it is. A reflection here is an operation of determinant minus one whose fixed set is a plane; an inversion centre has determinant minus one and fixes only the origin and is not one. The classes with a free ring are exactly the classes generated by their reflections, which is Chevalley's theorem checked rather than quoted.

Twelve of the thirty-two are free

A crystal class leaves some polynomials alone, and the ones it leaves alone form a ring. For twelve of the thirty-two classes that ring is generated by three polynomials with no relation between them, and for the other twenty it is not — and the twelve are exactly the classes generated by their mirror planes. The two verdicts are computed by routes sharing no code, and an inversion centre is not a mirror.

point-groups · Invariants

Named alongside it

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

Point groupCartan integerCharacterChiralityCrystal classCrystallographic restrictionDihedral groupGlide reflectionGroupInvarianceInvariantLattice

All concepts