Crystallography & Symmetry
patterns derived from their groups
Concept

Transitive action — where it appears

An action of a group on a set in which any element can be carried to any other. A transitive action on n points is the same thing as a subgroup of index n, the stabiliser of one point, which is how colourings and subgroups are counted together.

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

Subgroups of index two, three and four. Every plane group with the number of subgroups it has at each small index, counted by enumerating the transitive actions on that many points. The zeros are the interesting entries: p3 has no subgroup of index two and the four-fold groups have none of index three, because a subgroup of index n gives an action on n points and the group has to have a quotient that can act. A rotation of order three has nowhere to go in a set of two, and one of order four has nowhere to go in a set of three that is not the identity — so the index is constrained by the point group before any geometry is done.

How many subgroups of index three

Taking operations away and closing what is left finds the maximal subgroups and stops there. Counting instead the ways a group can act on three points finds all of them — and finds that a four-fold group has none of index three at all.

operations · Subgroups 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

Named alongside it

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

Colour symmetryIndexSublatticeAbelianisationEnumerationNormal subgroupNormaliserSubgroupConjugacy classCosetDivisor sumEigenvalue

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