Two colours, and a symmetry that swaps them

A chessboard and a grid of identical squares are the same pattern as far as the seventeen are concerned. Both are p4m; the classification looks at where the shapes are and never at what colour they were painted. That is a deliberate restriction and it throws away the thing a person looking at a chessboard notices first.

Admitting colour costs one idea. A symmetry of a two-coloured pattern either leaves the colours alone or exchanges them, and exchanging them twice leaves them alone — so the colour rule is a map from the group onto the two-element group, respecting composition. Its kernel is the half that preserves the colours, and every colouring is therefore a subgroup of index two.

This is the classification decorators have used for centuries under the name counterchange, and it is one of the few places where the mathematics arrived long after the practice.

What a colour-reversing operation looks like #

The clearest case is the one everybody already knows.

Three features of that picture are worth naming because they recur in every case.

The colouring can have a bigger repeat than the pattern. If a translation is colour-reversing, the coloured design only comes back to itself after two steps. A chessboard’s shapes repeat every square; its colours repeat every two.

Half the operations reverse and half preserve. Always exactly half, because the colour-preserving subgroup has index two, and a coset is the same size as the subgroup it came from. The figure asserts the equal split rather than stating it.

The two colours occur equally often. A colour-reversing operation is a bijection from the black points to the white ones, so their counts are equal. That is also asserted while the figure is drawn, and it is the kind of claim that would fail loudly if the colour rule were being applied inconsistently.

Counting them, group by group #

The enumeration behind the first figure is small and blunt. A homomorphism onto the two-element group kills every square, and the square of a translation is a translation by twice as much — so the whole question lives in the group modulo double translations, which is finite and has at most forty-eight elements. Every assignment of colours to a generating set is then tried, propagated over that finite group by closure, and kept if it never assigns two values to one element.

The counts that come out are 3, 7, 15 and 1 — one less than a power of two every time, which is what a count of non-trivial homomorphisms onto a group of order two has to be. That is asserted rather than admired: a count of, say, five would mean the enumeration had gone wrong somewhere, and there is no way to arrive at it legitimately.

pmm admits the most, with fifteen. It is generated by two mirrors and two translations, each of which can independently be made colour-reversing or not, giving sixteen assignments of which one is the trivial colouring.

Several admit exactly one. p6, p3m1 and p31m each have a single two-colouring. In p6’s case it is the one where the sixfold rotation itself reverses colour — a design that comes back to its own colours only after a third of a turn.

The group that cannot be two-coloured #

p3 admits none, and the reason is short enough to give in full.

A homomorphism onto a group of order two sends every element of odd order to the identity, because the image of an element of order three has order dividing both three and two. So the threefold rotation is killed immediately.

The translations go the same way, and that is the part worth spelling out. In the abelianised group a translation and its image under the rotation are indistinguishable, and the map sending a translation to itself minus its rotated image is three-to-one on the lattice — so every translation is three times something, and its image in any abelian quotient has order dividing three. Nothing is left that could be sent to the non-trivial element, and no surjective homomorphism exists.

A pattern with p3 symmetry cannot be painted in two colours so that its symmetries survive. Not “is difficult to”, and not “has not been done”: there is no such colouring, and the enumeration finds none because none is there.

That is a genuinely surprising constraint on decoration, and it is checkable by hand on any threefold design. Take a p3 pattern, colour one motif black, and follow the rotation round: the three images of that motif are related by rotations of order three, so they must all have the same colour, and the constraint propagates through the whole plane until everything is one colour.

The exception in appearance is the familiar hexagonal design in three colours. Three-colour symmetry is a different classification — homomorphisms onto the three-element group rather than the two — and p3 is rich in those. What it lacks is specifically the swap.

Which half keeps the colour #

A colouring is named by its colour-preserving subgroup, and looking at which subgroup it is turns the count into something more informative than a number.

For p4m, of order eight, each colouring leaves a subgroup of order four behind: p4 if the reversing operations are the reflections, and one of the mirror groups if the reversing operations include the rotation. So the seven colourings of p4m are seven different ways of writing a familiar group as the colour-preserving half of a familiar larger one, and every entry in the classical black-and-white tables is a pair of ordinary group names for that reason — p4m′ and its relatives are names of pairs, not of new groups.

That also explains why the numbers here are so uneven. A group with many index-two subgroups has many colourings; a group with few has few; and the count is not related to the group’s order in any simple way. pmm, of order four, has fifteen. p4m, of order eight, has seven. p6m, of order twelve, has three. Larger groups are more constrained, not less, because more of their operations are forced to agree.

Reading a counterchange design #

Put the other way round, the classification is a procedure for looking at a black-and-white pattern.

First find the group of the shapes, ignoring colour entirely, which is the ordinary classification the rest of this site is about. Then ask which of those operations exchange the colours, and check that the answer is consistent — half of them must, and the ones that do not must form a group.

The consistency check is where a design either is or is not a two-colour pattern. A colouring that reverses at some places and not at others, with no rule behind it, is decoration rather than symmetry, and no amount of pattern-matching will make it one of the classes.

That check is exactly what the enumeration performs, and it is the reason a rule that fails is discarded rather than repaired: an assignment of colours that is inconsistent on one element of the group is inconsistent as a design, and the figure that would have drawn it refuses instead.

Where the number forty-six comes from #

The standard reference count for two-colour plane groups is forty-six, and the enumeration here gives seventy-four. Both are right, and the difference is exactly the kind of thing this site is careful about.

Seventy-four is the number of index-two subgroups, summed over the seventeen groups. Forty-six is the number of equivalence classes of those subgroups, where two colourings count as the same when a change of basis turns one into the other.

The clearest instance is p1, which has three two-colourings: alternate along one axis, along the other, or along the diagonal. As subgroups these are genuinely different. As designs they are the same design seen with a different pair of axes chosen — and a classification of patterns should not distinguish them, since which pair of translations is called the basis is the same kind of choice as which cell is drawn.

This site does not compute the reduction. Deciding when two subgroups are related by a change of basis means working with the affine normaliser of each group, which is not something the integer machinery here does, so forty-six is quoted from the literature and seventy-four is enumerated. Reporting the enumerated number as if it were the classical one would be the more comfortable option and would be wrong twice over — once about the number and once about what was computed.

What this says about ornament #

The connection to decorative practice is real and it runs in a particular direction.

Counterchange is one of the oldest devices in pattern-making — black and white tiles, alternating tesserae, a border where the figure and ground exchange along its length — and it long predates any of the mathematics. What the classification adds is not a technique but a census: it says how many essentially different counterchange schemes a given layout admits, and in one case that the answer is none.

That bears directly on the survey question. Analysts disagreeing about which groups appear in a corpus disagree in large part about whether colour counts, and the disagreement is not resolvable by looking harder. A panel of shapes with p6m symmetry whose colouring breaks half of it is p6m if colour is ignored and p3 or p31m if it is not, and both readings describe the same tiles. The two-colour classification is where that question has a definite answer, because it names the colouring as part of the object rather than as a property of the object’s presentation.

Why the counts are one less than a power of two #

Every count in the table has the form 1, 3, 7 or 15, and the pattern is not a coincidence of small numbers.

The homomorphisms from a group onto the two-element group, together with the trivial one that sends everything to zero, form a vector space over the field of two elements: adding two colour rules pointwise gives another valid colour rule, since the sum of two homomorphisms into an abelian group is a homomorphism. A vector space of dimension r over that field has exactly 2ʳ elements, one of which is trivial, so the number of genuine two-colourings is 2ʳ − 1.

The dimension r is the number of independent choices available — how many generators can be assigned a colour value freely once the relations are respected. For pmm the answer is four and the count is fifteen; for p4m it is three and the count is seven; for p6 it is one; for p3 it is zero.

That structure is what the figure asserts rather than admires. A count of five or six would be impossible, so measuring one would mean the enumeration had gone wrong, and the assertion is the cheapest possible check on a computation whose answer nobody knows in advance.

It also explains why the colourings of a group are related to each other rather than being a bag of unrelated designs. Two colourings can be added, and their sum is a third — which in the picture means overlaying two counterchange schemes and taking the colour to be reversed where exactly one of them reverses it. The seven colourings of p4m are the seven non-zero vectors of a three-dimensional space over two elements, and adding any two of them gives a third one on the list.

The frieze case, which can be done by hand #

The strip’s version of the same question is small enough to check without any machinery, and it is the right place to be convinced.

A frieze group has at most four operations beyond translation, and a two-colouring assigns each of them a colour value consistently. p11m — a horizontal mirror and nothing else — has three colourings: the mirror reverses, the translation reverses, or both do. p1, with only the translation, has one. And the pattern of counts is the same as in the plane, for the same reason: it is the number of non-zero vectors in a space over the field of two elements.

The one worth drawing by hand is p11m with a colour-reversing translation, because it is a familiar object. Alternate motifs black and white along a strip, each reflected in the centre line: a row of alternating light and dark tiles. It has all of p11m’s symmetries, half of them exchange the colours, and it is the border every tiled floor in Europe has somewhere.

Where the exactness stops #

Two colours only. Three-colour and n-colour symmetries are the homomorphisms onto larger groups, and the counting is correspondingly larger. Nothing here bears on them, and p3’s poverty in two-colourings is matched by a richness in three that this page does not compute.

Colourings, not designs. What is enumerated is the rule saying which operations swap the colours. Turning a rule into a picture needs a motif, and a motif at the wrong place makes the colouring inconsistent — a point fixed by a colour-reversing operation would have to be both colours. The drawing code refuses that outright rather than picking one, which is the two-colour form of the comma rule.

The quotient is the object counted. Everything is computed in the group modulo double translations, which is legitimate because a homomorphism onto a group of order two cannot see anything finer. It also means the counts here say nothing about colourings with a period longer than two cells, and those exist for other reasons and other colour groups.

The same forty-six, under another name #

The classification counted here has a second life in a subject with nothing decorative about it, and the two arrived independently about twenty-five years apart.

Heesch introduced antisymmetry in 1929: a group with one extra operation of order two attached, commuting with everything, and no application in mind. That is exactly the structure a two-colouring has — the colour swap is an operation of order two that commutes with every motion, since swapping and then moving is the same as moving and then swapping.

So the two-colour plane groups are the antisymmetric plane groups, and the census reads the same way in either vocabulary. Seventeen ordinary groups, in which nothing swaps. Seventeen grey groups, in which the swap is a symmetry on its own — which is a pattern where every point is both colours at once, and is what an unpainted design amounts to. And forty-six black-and-white groups, in which some operations swap and others do not. Eighty in total, and that is the standard count of magnetic plane groups.

The subject that needed them is magnetism, where the two colours are the two senses of a magnetic moment and the swap is the reversal of time. An antiferromagnet is a crystal whose moments alternate, so a translation that carries one sublattice onto the other reverses every moment — precisely a colour-reversing translation, with the enlarged repeat this essay’s first figure shows.

The arithmetic transfers unchanged because it never mentioned colour. What is counted is homomorphisms onto a group of order two, and the two elements of that group can be black and white, up and down, or the presence and absence of a prime on a symbol. The p3 result transfers with it: a three-fold pattern admits no consistent two-colouring, so a class of odd order has no black-and-white descendant either, for the same divisibility reason and with no new argument required.

Where the ladder goes next #

The measurement of which groups decoration actually uses, and why the surveys disagree, is the groups ornament actually uses.

The famous case where the disagreement is at its sharpest, and where colour is the largest part of it, is the Alhambra question.

The subgroup arithmetic that a colouring is an instance of — index two, half the operations, twice the domain — is domains of a subgroup.

What the pictures here cannot show. Each drawing is one colouring of one group, and the interesting claims are about the set of colourings — that pmm has fifteen and p3 has none. No picture shows an absence, and the figure that carries the counts is a bar chart of a computation rather than a picture of any pattern at all.