Crystallography & Symmetry
patterns derived from their groups
Series

Colour — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · classification
  2. How much a count of descriptions over-counts. For each plane group that has any two-colouring at all: how many colourings it has, how many designs those come to, and the ratio between them. Over the seventeen the ratio is 1.61, and group by group it runs from 1.00 — where nothing is identified — to 3.50 at p2, whose seven colourings fall into one class of six and one of one. The tick on each row is that row's largest single class, and it is at least the bar and usually more. The largest class anywhere is p2's 6, and that same group over-counts by only 3.50, because a factor is a mean over the group's classes and a mean reaches its largest term only when every term equals it. Reading the largest class as the over-count is therefore an over-statement, always. And the factor varies from group to group, which is why no single correction turns a count of descriptions into a count of designs after the fact.

    Seventy-four colourings, forty-six groups

    This site counts the two-colourings of the seventeen and gets seventy-four. The literature says there are forty-six two-colour wallpaper groups. Both numbers are right, and the gap between them is a disagreement about when two coloured patterns are the same pattern.

    part 2 · classification
  3. 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 — that is the first rung of this ladder and it 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.

    part 3 · classification

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