Crystallography & Symmetry
patterns derived from their groups
Generator

Seventy-four homomorphisms, forty-six groups

Seventy-four homomorphisms, forty-six groups
Seventy-four homomorphisms, forty-six groups. Each row is a plane group: how many two-colourings it has, and how many of those are genuinely different designs. The first count is of homomorphisms onto the two-element group and comes to 74; the second identifies colourings related by an affine map carrying the group onto itself, and comes to 46 — the classical number of two-colour wallpaper groups. pmm supplies the largest gap, with fifteen colourings and five designs. Both numbers are computed here and neither is looked up.

Each row is a plane group: how many two-colourings it has, and how many of those are genuinely different designs. The first count is of homomorphisms onto the two-element group and comes to 74; the second identifies colourings related by an affine map carrying the group onto itself, and comes to 46 — the classical number of two-colour wallpaper groups. pmm supplies the largest gap, with fifteen colourings and five designs. Both numbers are computed here and neither is looked up.

4 essays call colour-classes. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

An antiphase boundary in p4. Where two antiphase states meet. Above the line the species alternate one way and below it the other, so at the boundary two cells of the same species sit next to one another and the ordering is out of step. This is a domain wall with no change of orientation across it: the crystal is not twinned, its lattice is undisturbed, and diffraction sees it only in the width of the superlattice reflections. Symmetry at work

The domains a lost translation makes, which nothing optical can see

An ordering transition can leave the crystal class untouched and take away translations instead. The domains that result have the same orientation, the same shape and the same optical properties as each other, and where two of them meet the ordering is simply out of step — a boundary with no change of direction across it and no way to find it except by looking at the ordering itself.

 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. The classification

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.

 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. The classification

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.

 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. The classification

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.

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