Crystallography & Symmetry
patterns derived from their groups
Concept

Colour symmetry — where it appears

Symmetry in which an operation may permute colours as well as move points, so the group comes with a homomorphism onto those permutations. It gives a finer classification of patterns than the seventeen, and one of the seventeen admits no two-colouring at all.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

Two-colourings of the seventeen. How many ways each of these 17 plane groups can be two-coloured so that every symmetry either preserves the colours or exchanges them. 74 in all, each one a subgroup of index two enumerated by trying every assignment of colours to a generating set and keeping the assignments that turn out to be consistent. p3 admits none: a homomorphism onto a group of order two has nothing to send a three-fold rotation to but the identity, and once the rotation and its conjugates are killed nothing is left to reverse the colours. pmm admits the most, with 15. Every count is one less than a power of two because the homomorphisms of a group onto the two-element group are the non-zero elements of a vector space over that field.

Two colours, and a symmetry that swaps them

A chessboard and a grid of identical squares have the same group, which is plainly not what anybody sees. Admitting the colour swap as an operation gives a finer classification — and one of the seventeen turns out to admit no two-colouring at all.

classification · Ornament How often a dot gives the wrong group. Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it.

The Alhambra question

Textbooks say the Alhambra contains all seventeen wallpaper groups. Careful analysts of the same building have counted eleven, thirteen, fourteen and seventeen — and the disagreement is not about the mathematics but about what counts as an instance.

classification · Ornament Subgroups of index 3, across the seventeen. Every subgroup of index 3 with cyclic quotient in each of the seventeen plane groups, sorted into the two kinds: 4 keep all the translations and lose operations, 22 keep all the operations and lose translations, and the total is 26. The split is decided by whether the homomorphism onto ℤ3 kills the two lattice translations, which is a property of the kernel and not a judgement. Every one of them is found by enumeration inside the finite quotient by 3Λ, and the count for the whole classification is a measurement.

Three colours, and why most patterns cannot have them

Seventy-four of the seventeen plane groups' subgroups have index two, and every group but one has at least one. At index three there are twenty-six, and ten of the seventeen have none at all — because a symmetry of order two cannot survive being asked to permute three colours.

classification · Colour What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Every colour count at once

Eight hundred and five structures is the answer for two species on sixteen sites. For three species it is a different sum, and for four another. Averaging cycle counts instead of fixed-point counts turns the answer into a polynomial — and refining the same average says how many structures there are at each composition, which is the number anybody actually needs.

operations · Counting One group refuses two colours and three refuse three. The two counts side by side, with the rows that refuse a number of colours marked. p3 is the only group with no two-colouring; p4, p4m and p4g are the only ones with no three-colouring. Neither list is a subset of the other and both come from the same arithmetic — a rotation order that divides nothing the symmetric group has.

What a half-turn does to three colours

Ten of the seventeen plane groups have no three-colouring, because a half-turn cannot permute three colours cyclically, and that is true. Drop the word cyclically and the answer changes completely: a half-turn permutes three colours perfectly well by swapping two and fixing one, and only the three four-fold groups refuse three colours at all.

classification · Colour A structure that is its own negative, and one that is not. Two arrangements of two species on a four-by-four block with p4m symmetry, each with eight sites of each species, beside the arrangement that exchanging the species produces. On the left the exchanged arrangement is the original one moved by an operation of the group, so the two are one structure: it is self-complementary, and its balance of species is forced by a symmetry. On the right the exchanged arrangement is a genuinely different structure, and the balance is a coincidence of count. Counting structures with the species interchangeable merges the pair on the right into one and leaves the one on the left as it was, which is why the count does not halve.

Swapping the species does not halve the count

Let the two species on a lattice be interchangeable and every structure pairs with its negative, so the count should halve. It does not, because some structures are their own negative: an operation of the group carries them onto the arrangement with the species exchanged. Those are counted by the operations whose every cycle is even, they all sit at exactly half composition, and read as spins they are the arrangements whose zero magnetisation a symmetry enforces.

operations · Counting Eighty layer groups, over the seventeen plane groups. Each of the seventeen plane groups with a block for every layer group that projects onto it. Every plane group carries one layer whose two faces differ and nothing turns it over, and one with a mirror in its own plane. The rest are turned over by some of their operations, and there is one for each class of two-colouring of the plane group: the pale-dark blocks turn the layer over only with a rotation, a mirror line or a glide line of the pattern, and the dark blocks turn it over on a translation — a glide plane lying in the layer. The totals are 17, 17, 26 and 20, which is eighty.

Eighty is seventeen, seventeen and forty-six

The eighty layer groups are usually quoted, and an earlier count here reached sixty-three and blamed the translations. The eighty follow in three lines from things already counted: a layer group is a plane group with a sign on each operation, and the signs are either all plus, all doubled by a mirror in the layer, or a two-colouring of the plane group — seventeen, seventeen and forty-six. The sixty-three had missed the twenty colourings that turn a layer over on a translation, which are glide planes lying in the layer itself.

classification · Subperiodic Every four-colouring of the seventeen, by the group that permutes the colours. The 281 subgroups of index four across the seventeen plane groups, each a colouring of a pattern in four colours that the pattern's symmetries permute transitively, sorted by the permutation group they induce on the colours: ℤ₄ 20, Klein 77, D₄ 164, A₄ 8, S₄ 12. The groups without a three-fold rotation use only the cyclic, Klein and dihedral groups; the five with one use only A₄ and S₄, apart from a single Klein colouring of p6m.

Four colours, five ways to permute them

Two colours can be swapped in one way and three can be permuted in two. Four colours can be permuted transitively by five different groups, and the seventeen plane groups use all five: 281 four-colourings sort into 20 cyclic, 77 Klein, 164 square, 8 tetrahedral and 12 fully symmetric. The last twenty belong to the five groups with a three-fold rotation, and they are the lattice reduced modulo two.

classification · Colour Ninety-six four-colour designs. For each of the seventeen plane groups, the number of its four-colourings, as subgroups of index four, and the number of distinct designs they fall into under the affine normaliser, sorted by the group that permutes the four colours. The 281 colourings make 96 designs: 13 ℤ₄, 39 Klein, 39 D₄, 2 A₄, 3 S₄.

Ninety-six four-colour designs

The seventeen plane groups have 281 four-colourings, counted as subgroups of index four, and many of them are the same design described from a different origin, on different axes or from a neighbouring cell. Identified under every affine map that carries the pattern's group to itself, they make ninety-six designs. The same computation run at two and three colours gives forty-six and twenty-three, and all three numbers match the classical counts of coloured plane groups.

classification · Colour 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.

EnumerationIndexHomomorphismNormaliserOrbitSublatticeCounterchangeSubgroupTransitive actionBurnsideEisenstein integerGaussian integer

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