Crystallography & Symmetry
patterns derived from their groups
Concept

Sum of two squares — where it appears

A whole number expressible as a squared plus b squared, which is exactly when the square lattice has a square sublattice of that index. Which superstructures a surface can form is therefore decided by a theorem of Fermat's about primes.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

Which indices have a square sublattice. For each index up to 26: how many sublattices of the square lattice are themselves square, found by testing whether the quarter-turn maps each one onto itself; the same count as a sum over divisors, +1 for each divisor one more than a multiple of four and −1 for each one less; and the ways of writing the index as a sum of two squares. The three agree at every row, which is Fermat's theorem — and it says that 3, 7 and 11 have no square sublattice at all while 5, 13 and 17 have two.

The sublattices that stay square

A sublattice of the square lattice is itself square exactly when its index is a sum of two squares — so index five has two and index seven has none, and which superstructures a surface can form is decided by a theorem of Fermat's about primes.

lattices · Sublattices Every coincidence index is odd. The rotations that bring a cubic lattice into coincidence with itself, up to index 25: 17 distinct relations across 12 indices, each found by enumerating integer quaternions and each index computed as the size of a sublattice rather than from the usual formula — the two are then required to agree. 5 of the 12 indices carry more than one relation, which is why the tables write 13a and 13b. Every index is odd. That is not a feature of this range: a rational orthogonal matrix written in lowest terms has an odd denominator, so the factors of two always cancel.

Every coincidence index is odd, and in the plane most of them do not exist

The indices at which two copies of a cubic lattice share points are 3, 5, 7, 9, 11 and every odd number after them. There is a two-line proof that no even index can occur. Ask the same question about a square lattice and the answer is a different list entirely, governed by which numbers are sums of two squares — so Σ3, which is the commonest boundary in every metal, has no plane analogue at all.

applied · Interfaces The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry.

How many vectors of each length

Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven.

lattices · Lengths At which indices a group contains a copy of itself. A filled circle where the group has a subgroup of that index which is the same plane group again. The groups with no rotation past a half-turn take every index — the lattice can be stretched along one direction by any factor. The four-fold groups take the sums of two squares and the three- and six-fold groups take the Loeschian numbers, because a sublattice invariant under a quarter or a third of a turn is an ideal in the Gaussian or Eisenstein integers and its index is a norm. The groups with mirrors take fewer still, and p4g takes only the squares.

The same group in a bigger cell

A subgroup usually gives something up. An isomorphic subgroup gives up nothing but scale — the same plane group again, on a coarser lattice — and the indices at which that is possible turn out to be the values of a quadratic form.

space-groups · Isomorphic subgroups The turns a rolled sheet can make are square roots. Every primitive rolling of a square and of a hexagonal sheet with at most 66 cells in a repeat, one mark per distinct turn. Across is the number of cells n₀ in one repeat of the tube; up is the turn, as a fraction of a circle, of the screw that climbs one n₀-th of the repeat. Most columns are empty, and the occupied ones hold two or four marks rather than anything up to n₀: on the square sheet the turns are exactly the square roots of minus one modulo n₀, on the hexagonal sheet the roots of the corresponding congruence for minus three. The pattern is symmetric about half a turn because a root and its negative are the two hands of one tube.

A rolled sheet turns by a square root

Roll a square sheet into a tube of thirteen cells and its screw could, on paper, turn by any of twelve fractions of a circle. It turns by five thirteenths or eight, and nothing else, because five squared plus one is a multiple of thirteen. Every rolled tube's screw is a square root of minus the sheet's determinant — which is why handed tubes appear only at the orders Fermat allows, why they always come in both hands, and why the crystallographic ones need an accident.

restriction · Finite groups 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 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.

Gaussian integerIndexSublatticeColour symmetryEisenstein integerEnumerationLoeschian numberNormaliserQuadratic formSuperstructureClass numberCoincidence site lattice

All concepts