Crystallography & Symmetry
patterns derived from their groups
Concept

Dihedral group — where it appears

The group of an n-fold rotation together with the reflections it generates, of order 2n. Its invariant ring is free with generators of degrees two and n, for every n whether or not a lattice permits that rotation.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Every solution of the axis equation. The integer solutions of 2 − 2/N = Σ(1 − 1/nᵢ), which is what counting the pairs (rotation, fixed pole) two ways gives. Two classes of axis force n₁ = n₂ = N and give the cyclic groups; three classes give the dihedral family and exactly three sporadic answers — (2, 3, 3), (2, 3, 4) and (2, 3, 5), of orders 12, 24 and 60, which are the rotation groups of the tetrahedron, the octahedron and the icosahedron. Four classes are impossible, because four terms of at least a half already exceed the left-hand side. Nothing about crystals has been used.

Before the lattice has a say

Every finite group of motions of the plane is a Cₙ or a Dₙ, and every finite group of rotations of space is one of five families. Both lists come out of counting rather than out of crystallography — and then the crystallographic restriction deletes almost all of them, leaving eleven.

restriction · Finite groups The invariant degrees exist for every n; the lattice permits five of them. The reflection group with an n-fold rotation has an invariant ring generated in degrees 2 and n, for every n whatever — the dimensions on the right are counted by pairing monomials in complex coordinates, which needs no matrix and therefore no lattice. Five of these groups can be written in integer matrices, and those five are named in the middle column; the rest cannot, because a lattice has no five-fold or seven-fold rotation. The crystallographic restriction is usually a statement about traces of matrices. Here it is the statement that only five of these invariant rings belong to a crystal, and the two arguments have nothing in common but their answer.

The degrees that name the restriction

The reflection group with an n-fold rotation has invariants of degrees 2 and n — for every n, with no lattice anywhere in the argument. Which of those groups a crystal may have is then the only question left, and its answer is the crystallographic restriction arriving from a direction nobody points it from.

restriction · Restriction 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 Which two kinds of operation make each of the 64 two-generator space groups. For each of the fifteen pairs of kinds of operation, rotation, screw, mirror, glide and rotoinversion (the inversion included), the number of the 64 space-group types generated by two of their own operations that some pair of those kinds generates. Every one of the 960 cases is decided: by a generating pair found, by the pair's linear parts failing to generate the point group, or by a proof in a finite quotient of the group. A screw and a glide make 32 types; two mirrors and a rotation with a mirror make none.

Two mirrors never make a crystal

Sixty-four space groups are generated by two of their own operations. Which two kinds can do it is a finite question, and it has been settled for all sixty-four and all fifteen pairs of kinds. A screw and a glide make thirty-two of them. Two mirrors make none, and neither does a rotation with a mirror, for reasons that need no computation: two operations that undo themselves generate a dihedral group, and an axis and a plane always share a point or a direction. Two plain rotations make six groups, every one of them cubic and without a mirror.

space-groups · Screws and glides

Named alongside it

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

Crystallographic restrictionMirrorCartan integerCyclic groupCyclotomic polynomialsFinite groupGenerating setGlide planeIcosahedral symmetryInvariant degreesInvariant ringLattice

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