Seventy-four colourings, forty-six groups

Two colours and a swap sets the machinery up: a two-coloured pattern is an ordinary plane group together with a rule saying which of its operations exchange black and white, that rule is a homomorphism onto the two-element group, and its kernel is a subgroup of index two. Enumerating them is a finite calculation — the whole question lives in the quotient by double translations — and running it across the seventeen gives seventy-four.

Every textbook of colour symmetry says there are forty-six two-colour wallpaper groups.

Neither number is an error. They count different things, and the difference between them is the most useful thing in this corner of the subject: seventy-four counts homomorphisms of one fixed group; forty-six counts them up to the moves that carry the group onto itself.

What is being identified #

The move that identifies two colourings is a change of description, and it is exactly the object the same pattern described twice is about: the normaliser, the set of motions carrying a plane group onto itself.

An element of the normaliser maps the group’s operations to the group’s operations, so it maps a homomorphism to a homomorphism — send φ to φ∘α⁻¹ — and two colourings related that way are the same design seen from a different origin, or along a different pair of axes. Nobody looking at the two patterns would call them different, and no property of either distinguishes them.

pmm is the extreme case and the clearest. It has fifteen two-colourings. Five of them are genuinely different designs; the other ten are those five drawn from another origin. The orbits have sizes 1, 2, 4, 4 and 4, and the ones of size four are single designs wearing four descriptions apiece.

The normaliser here is affine, and that matters #

There is a choice in what counts as a change of description, and it is the choice that decides the number.

A Euclidean normaliser admits only isometries: rotations, reflections and translations that carry the group onto itself. An affine normaliser admits anything linear that does — including shears, which are not isometries and which nevertheless carry a rectangular lattice onto itself and pm onto pm.

The classification of coloured patterns is affine, and it has to be. Two counterchange designs that differ only by a shear of the underlying lattice are the same design in every sense a designer or a crystallographer uses: the same operations preserve the colours, the same ones reverse them, the same repeat structure holds. Insisting on isometries would split classes on a distinction nothing depends on.

So the group searched here is every map α(x) = Ax + a with A an integer matrix of determinant ±1 and αGα⁻¹ = G — a strictly larger group than the Euclidean normaliser the site computed for the origins essay, and the right one for this question. Where the two differ, the affine one merges more and the count is smaller.

Where the seventy-four sit #

Reading the census down the column of colourings gives a pattern that is worth stating because it is arithmetic rather than taxonomy.

Every count of colourings is one less than a power of two. That is not a coincidence: the colourings of G are its surjective homomorphisms onto ℤ₂, which are the non-zero elements of a vector space over the two-element field — the space of homomorphisms — so there are 2ʳ − 1 of them, where r is the dimension. pmm has fifteen, so r = 4; p1 has three, so r = 2; p6 has one, so r = 1.

Exactly one group has none. p3 admits no two-colouring at all, and the reason is short: a homomorphism onto a group of order two must send an element of order three to an element whose order divides both three and two, which is the identity. Kill the three-fold rotation and its conjugates and there is nothing left to reverse the colours. Three colours and why most cannot is the other side of the same argument, where three-fold groups are the ones that can and the rest cannot.

Where the forty-six sit #

The counts after the identification are less tidy, and the untidiness is the content.

p1 has three colourings and one design. The three index-two subgroups of the translation lattice are the three sublattices of index two, and a change of basis carries any of them to any other — so all three are the same black-and-white pattern, the one with stripes.

p2 has seven and two, and it is the extreme case in the other direction from cm. Six of its seven colourings are one design: the affine normaliser of p2 is large — every integer change of basis carries p2 to p2, because a half turn is a half turn in any basis — and it sweeps six of the seven index-two subgroups onto one another. The seventh is the one it cannot move.

cm has three and three: nothing is identified, because cm’s affine normaliser is small. A centred rectangular lattice fixes both of its axis directions — one is the mirror, the other is perpendicular to it — so almost no change of basis survives, and each of the three colourings stands alone.

p4g has three and three; p4m has seven and five. Two groups of the same order, with the same number of subgroups of index two in one case and different in the other, and different fractions of them identified. There is no formula here, which is why the count is a computation.

The total is 46, and it is the classical number — the forty-six two-colour wallpaper groups that Woods enumerated in 1936 and that appear in every treatment of counterchange symmetry since. It is reached here without a table: seventy-four homomorphisms found by closure, an affine normaliser found by search, and the orbits counted.

The check a search owes #

The conjugating map is looked for by sweeping integer matrices with small entries and translations on a grid. A sweep that is too small cannot merge two colourings that a larger matrix would identify, so it reports too many classes — and every extra class looks like a discovery rather than like a bug.

That failure is silent and one-directional, so the count is computed at more than one bound and required not to move. It is forty-six with matrices to ±2 and to ±3, and with translations in quarters and in eighths. This is the check every search on this site owes and the reason the number can be stated rather than offered.

What the two counts are each good for #

Both numbers answer real questions and they are not interchangeable.

Seventy-four is the count a calculation wants. Enumerating the ways a structure can order antiferromagnetically on a given lattice, or the ways a two-species alloy can decorate a given group, is a count of homomorphisms with the group fixed — the crystal has one setting and the descriptions are not interchangeable, because the atoms are somewhere in particular.

Forty-six is the count a classification wants. Asking how many different counterchange designs there are, or how many black-and-white patterns a textile can realise, is a question about designs and not about descriptions.

Confusing them produces a specific error, and it is worth naming: counting descriptions where designs were meant over-counts by a factor that varies from one to three and a half between groups. It is not a constant, so no correction factor fixes it, and a count made the wrong way cannot be repaired afterwards.

The four in that sentence came from somewhere and it is worth saying where, because the mistake is a natural one. The largest single class anywhere is p2’s six: six of its seven colourings are one design, seen from six choices of basis. The largest factor is also p2’s, and it is 3.50 — the mean of six and one. A factor is a mean over a group’s classes, and a mean reaches its largest term only when every term equals it, so a group whose classes differ in size always over-counts by strictly less than its worst class. Reading the worst class as the over-count is therefore an over-statement, always. pmm is the case that produces the four: its fifteen colourings fall into classes of 1, 2, 4, 4 and 4, so its largest class is four and its factor is 3.00. The figure above draws both — the bar is the factor and the tick that row’s largest class — and each row is refused unless the two stand in that relation.

The drawable colourings, and the ones that are not #

A colouring is a rule about operations, and drawing it needs one more thing to be true: the colour of a point has to be well defined.

A point’s colour is decided by the operation that produced it, and a point reached by two operations of opposite colour would have to be both at once. That happens exactly when the point’s stabiliser contains a colour-reversing operation — a motif sitting on a mirror that reverses the colours, say — and such a colouring cannot be drawn with that motif at all.

This is the two-colour form of the hazard the whole site is about, and it is sharper than the uncoloured version. A motif in the wrong place does not merely give the pattern a symmetry nobody claimed; it makes the colouring inconsistent, and nothing about the resulting picture announces which of the two things went wrong. So the figures here place the motif at a general position and check the consistency while drawing, and a colouring that cannot be drawn at that position is reported rather than drawn wrongly.

None of this affects the count. Whether a colouring can be drawn with a particular motif is a fact about the motif; the classification is of the homomorphisms, and all seventy-four exist whether or not a given comma can display them.

Where the exactness stops #

The enumeration is exact. The homomorphisms are found by propagating an assignment over a finite quotient and discarding the inconsistent ones, and consistency is an equality of values rather than a comparison of numbers.

The equivalence is exact given the search bound, and the bound is checked rather than assumed. What could still be wrong is a conjugator with entries larger than three, which no bound tested here would find; the stability of the answer across bounds is evidence and not proof.

Nothing here says which colourings a material realises. A magnetic structure that reverses spin under half its operations is one of these colourings, and which one it adopts is decided by exchange energies this site does not compute. The classification says what is available. Which magnetism a class permits is the same distinction on the point-group side, where the counting is of permissions rather than of arrangements.

Reading a design and finding its group #

The classification is only useful if a pattern can be placed in it, and placing one is a two-stage question that is easy to run in the wrong order.

First the uncoloured group. Ignore the colours entirely and find the plane group of the pattern as a set of shapes. That is the ordinary detection this site does everywhere, and it gives the group G.

Then the homomorphism. For each operation of G, does it preserve the colours or exchange them? The answer is a value in ℤ₂ for each operation, and it is a homomorphism because composing two colour-reversals preserves. That assignment is the colouring, and comparing it with the enumerated list places the design.

Running it the other way round — trying to find the coloured group directly — fails on the commonest case: a design whose colour-preserving half is itself a recognisable plane group can be mistaken for that group with decoration, and the operations that reverse the colours are then invisible. The colour-preserving subgroup of a p4m colouring can be cmm, and a reader who stops there has found a real group and the wrong answer.

Who counted them, and what they were counting #

The subject has a tangled naming history, and the tangle is a version of the same confusion.

H. J. Woods enumerated the forty-six in 1935–36, in the Journal of the Textile Institute, as part of a four-part study of the geometry of pattern design — the count arrived in a textile journal because counterchange is a weaver’s problem before it is a crystallographer’s. Heesch had described the idea in 1929, and Shubnikov’s systematic treatment in the 1940s and 1950s gave the groups the name they usually carry now, Shubnikov groups, and connected them to magnetic structures — where the two “colours” are the two directions of a spin and the colour-reversing operations are the ones combined with time reversal.

The plane case is the small one. In three dimensions the same construction gives the 1,651 magnetic space groups, of which 1,191 are genuinely two-coloured, and those numbers are index-two subgroup counts of exactly the kind computed here.

What a designer and a crystallographer each mean by the same picture #

The two counts have a counterpart in what the two trades do with a coloured pattern, and it is worth one paragraph because it explains why the literature is split across two disciplines.

A designer asks which designs exist, and identifies anything a change of viewpoint relates. Woods counted forty-six in a textile journal for exactly that reason: a weaver choosing a counterchange draft cares about the design and not about which corner of it is called the origin.

A crystallographer asks which structures a given crystal admits, and cannot identify descriptions, because the crystal is in a particular orientation with atoms at particular coordinates. An antiferromagnet ordering on a fixed lattice has seventy-four available homomorphisms in the plane case, and which one it takes is a physical question with a definite answer.

So the same enumeration is done twice in two literatures with two different answers, and neither is careless. The number depends on the equivalence, and the equivalence depends on what the count is for — which is the most transferable thing in this essay and applies to every enumeration on this site.

Where the ladder goes next #

This rung establishes what the equivalence is and what it costs. Three rungs are visible above it.

More colours. Three-colourings are homomorphisms onto ℤ₃, and only the groups with three-fold rotations have any — which is the reverse of the two-colour case and is three colours and why most cannot. The general classification is of homomorphisms onto any permutation group, and the counts grow quickly.

Colour groups whose lattice changes. A colouring can double the repeat, so the black sublattice is a genuine sublattice of the pattern’s own — which is the connection to how many sublattices, and the reason a counterchange design needs a bigger cell than the pattern it colours.

The two kinds of subgroup, coloured, and it is a finer classification than the count. An index-two subgroup either keeps the lattice and drops half the point operations, or keeps all of them and halves the lattice — the translationengleiche and klassengleiche division of the two ways down. Both kinds appear among the seventy-four, and the two produce visibly different designs: the first swaps colours under a rotation or a mirror, the second under a translation, which is a stripe.

The split is twenty-nine and forty-five. Twenty-nine of the seventy-four colourings keep the whole lattice and forty-five halve it, and after the identification the forty-six break as twenty-six and twenty. That the second pair adds up at all is worth a sentence: the kind has to survive the identification for the question to make sense, and it does, because an affine normaliser element has an integer unimodular linear part and therefore carries the lattice onto itself. It cannot turn a colouring that keeps every translation into one that reverses some. If it ever did, the two splits would stop being splits of the same thing, and the figure above compares them and refuses to be drawn when they disagree.

Two of the rows are decided by the point group alone, and the rest are not. A colouring of the first kind is a homomorphism that kills every translation, so it is a homomorphism of the point group onto the two-element group — and a group has one of those exactly when its point group has even order. p1 and p3 are the two whose point groups have odd order, and they are exactly the two rows with nothing in the filled column. Nothing so tidy decides the second kind: p4 has two colourings of it and p4g has none, on point groups of the same order, because whether a sublattice of index two can be the colour-preserving half depends on the glides as well as the rotations.

A three-fold rotation forbids the second kind outright, and that one is decidable. There are exactly three sublattices of index two in the plane, and a three-fold rotation permutes them in a cycle — so no single one of them is carried onto itself, and none can be the colour-preserving half. p3, p3m1, p31m, p6 and p6m therefore sit entirely in the filled column, which is the same arithmetic that gives p3 no colourings at all, applied to translations instead of to rotations.

The magnetic case. In three dimensions the same arithmetic is done with time reversal as the colour swap, and this site has the point-group half of it already. The space-group half is the 1,651, and the identification problem there is the one solved here with an affine normaliser — at a scale where getting it wrong is a difference of several hundred groups.

Forty-six, seventeen and seventeen #

The forty-six are usually quoted alone, and they are one part of a total that this collection has already met under another name.

Three things can happen to the colours. A plane group can act with no operation swapping them — every operation preserves colour, and the pattern is an ordinary uncoloured one with a colour painted on. It can act with some operations swapping and some not, which is the case the forty-six enumerate. Or the swap itself can be a symmetry: the operation that changes both colours and moves nothing.

Counting all three gives seventeen, forty-six and seventeen, and their sum is eighty.

Eighty is the number of layer groups, and the coincidence is not one. A layer group is a plane group in which every operation carries a sign saying whether it keeps the two sides of the layer or exchanges them — which is a homomorphism onto a two-element group, exactly like a colouring. The three cases correspond: a layer with no side-reversing operation, a layer whose reversals come attached to spatial operations, and a layer with a horizontal mirror, which reverses the sides and moves nothing in the plane.

So a layer is not a wallpaper and a two-colour pattern is not a wallpaper are the same statement, with “which side” in place of “which colour”. Anything proved about one transfers, and the eighty are one list serving two subjects.

The same count in space, and what it is for #

The three-dimensional version of this census is larger, older and has a use that the plane case does not.

There are 1,651 groups in the corresponding classification — 230 with no swap, 230 in which the swap alone is a symmetry, and 1,191 in which some operations swap and some do not. The last number is the direct analogue of the forty-six.

The colour is a spin. In a magnetic crystal each atom carries a magnetic moment, and an operation of the space group either preserves the moments or reverses them all. Reversing them is time reversal — the operation that turns every current backwards — so the two-element group being mapped onto is the group generated by it, and a magnetic structure is exactly a space group together with such a homomorphism.

The three cases are three kinds of magnet. No swap: every operation preserves the moments, which is the situation in a ferromagnet with the moments aligned. The swap alone a symmetry: reversing every moment changes nothing, which means there are no moments — a non-magnetic crystal. Some swapping and some not: an antiferromagnet, where the reversal is a symmetry only when combined with a spatial operation that exchanges the two sublattices.

And the classification is what a neutron experiment is interpreted against. Neutrons scatter from magnetic moments as well as from nuclei, so a magnetic ordering adds reflections to a diffraction pattern in the way an atomic ordering does — and deciding which magnetic structure produced them means deciding which of the 1,191 is present. That is the same problem as space-group determination, run on a classification seven times larger, and it is the reason these groups were worked out at all.