Crystallography & Symmetry
patterns derived from their groups
Concept

Eisenstein integer — where it appears

A number a + bω with a and b integers and ω a primitive cube root of one, so that the points of the hexagonal lattice form a ring. A prime splits into two such numbers exactly when it leaves remainder one on division by three, which decides the sublattices that keep a three-fold rotation.

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

Sums of two squares, arriving as superstructures. Which indices admit a sublattice of the same shape as the square lattice, drawn as a bar per index whose height is how many there are. The pattern is not a pattern about lattices at all: an index works exactly when it is a sum of two squares, because a similar sublattice of the square lattice is multiplication by a Gaussian integer and its index is that integer's norm. The indices that work up to 30 are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, and the same list is produced here a second time by factorising rather than by searching, with the two required to agree.

The sublattices that are the same shape

Thinning a lattice usually changes its shape. Sometimes it does not: the sublattice is the parent rotated and scaled, and a drawing of it alone would be a drawing of the parent. Which indices allow it turns out to be a question Fermat answered in 1640.

lattices · Sublattices Which plane groups can be coloured with five, seven, eleven or thirteen colours. For each of the seventeen plane groups and each of the primes 5, 7, 11 and 13, the number of its colourings with that many colours, as subgroups of that index, and the group that permutes the colours: cyclic (C), dihedral (D), or the affine group of the colours with a multiplier of order 3, 4 or 6 (written p:3, p:4, p:6). Totals 104, 172, 268 and 420. The square groups p4 appear at 5 and 13, the three-fold and six-fold groups p3 and p6 at 7 and 13, and p4m, p4g, p3m1, p31m and p6m never.

Five colours, seven colours, and the primes that split

Four colours gave the seventeen plane groups 281 colourings, five different groups permuting the colours and ninety-six designs. Five colours give 104 colourings, three groups and fourteen designs, and from five upwards every prime behaves the same way. A prime number of colours that the point group cannot divide forces every colouring onto a sublattice the rotations keep, and whether the square and hexagonal patterns have one is decided by whether the prime splits in the Gaussian or the Eisenstein integers.

classification · Colour Whole stars of the hexagonal zone, and the boxes that hold them. For ten wavevectors of the primitive hexagonal zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it differs, in the smallest n by n grid of cells holding the star. The corner K is a single arm up to sign and is held whole by three cells, against nine for a grid; K raised to the top face, H, by six against eighteen; halfway to K, T, by twelve against thirty-six. M needs two cells for one arm and four for its three, and L two and eight. Every smallest box for a whole star keeps the full hexagonal symmetry.

The corner of a hexagon fits in three cells

A periodic box for a hexagonal crystal holds a whole star of wavevectors only if it keeps the crystal's full symmetry, as a cubic box must. What the cube never offered is a box turned through thirty degrees. It holds the corner of the hexagonal zone in three cells where every grid needs nine, and the saving is always exactly a factor of three and always at the same stars. Turn a box through any other angle that the six-fold axis allows and it loses the mirrors, and comes with a mirror image.

classification · Flat space Which plane groups can be coloured with five, twenty-five, seven and forty-nine colours. For each of the seventeen plane groups, its colourings with 5, 25, 7 and 49 colours, as subgroups of that index. At p² colours the count splits into those whose translations meet the lattice in the whole plane pL and those that meet it in a cyclic sublattice. The five groups with a mirror across a rotation, shaded, have none at 5 or 7 and p² of them at 25 and 49, all on pL. In all, 104, 1449, 172 and 4197 colourings.

Twenty-five colours, and the mirrors come back

With a prime number of colours from five up, five of the seventeen plane groups can never be coloured at all: the ones with a mirror across a rotation. With the square of a prime they all can. A colouring with twenty-five colours may meet the lattice in the whole plane modulo five, which every group keeps, and the colours then form a plane the point group acts on by its matrices. The designs come out as the prime's count again, with one more for each of the seventeen groups.

classification · Colour

Named alongside it

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

SublatticeGaussian integerIndexColour symmetryEnumerationNormaliserSum of two squaresBrillouin zoneCommensurateEigenvalueMultiplicative functionQuadratic form

All concepts