Finitely many is not few

Why there is a list at all establishes that each dimension holds finitely many crystallographic groups, which is the third of Bieberbach’s theorems and the part of Hilbert’s eighteenth problem that asked for it. The proof is a chain: a divisibility bounds the order of the point group, a bounded order leaves finitely many groups of integer matrices, and finitely many ways of attaching translations to each.

Nothing in that chain says how many. A divisibility argument is indifferent to the size of the answer, and Minkowski’s bound does not know what a crystal is.

The answer is 2, 17, 230, 4,783, 222,018 and 28,927,922 for the first six dimensions, and the useful question is not why it is finite but where the length comes from. The classification is three steps — enumerate the finite groups of integer matrices acting on a lattice, attach translations to each in every consistent way, and quotient by the changes of basis that are relabellings — and the three do not contribute equally.

The plane is almost all first step #

The plane’s answer can be read off completely, and it is lopsided.

There are thirteen arithmetic classes, the same thirteen the search for finite groups of two-by-two integer matrices produces. Ten of them admit exactly one group: every consistent assignment of translations to them is removable by a change of origin, so the class carries its point group sitting on its lattice and nothing else. Three do not — the mirror on a rectangular lattice, 2mm on a rectangular lattice, and 4mm on a square one — and between them they supply seven of the seventeen.

So in the plane the second step is nearly idle. Thirteen classes become eighteen extension classes and then seventeen groups, which is a multiplier of 1.31, and most of that multiplier comes from one class. Almost everything about the number seventeen is decided by the search over matrix groups, and the cocycle arithmetic that makes the same count without a picture is mostly confirming that there is nothing to add.

That is the wrong lesson to take away, and taking it away is easy, because the plane is the case everybody learns.

The lopsidedness also explains something about how the seventeen are usually presented. A textbook derives them by listing the possible point groups and then noticing that a few of the mirrors can be glides, which is exactly the shape the numbers have: the first step does the work and the second is a short appendix. Nothing about that presentation survives into higher dimensions, and a reader who has met only the plane is left with an intuition that is wrong in the direction that matters most.

Not one of the six space classes carries a single group #

One dimension up the balance reverses, and the reversal is visible on a sample small enough to be enumerated from the operations rather than read.

The primitive monoclinic classes carry two each — P2 and P2₁, Pm and Pc. The primitive orthorhombic class with three two-fold axes carries four. The primitive four-fold class carries four and the primitive three-fold class three, which are the screw axes. And the primitive orthorhombic class with a polar axis carries ten, enumerated one candidate at a time from sixteen consistent assignments.

Not one of the six carries a single group, where ten of the plane’s thirteen do. The six average 4.17 groups a class, which is a sample and not a mean — the true figure over all seventy-three is 3.15 — but the direction is not in doubt, and it is the direction the whole column takes.

The reason is not hard to see once the numbers are side by side. An extension class is a way of attaching a fractional translation to each generator that no origin removes, and the room for that grows with two things at once: how many generators the point group needs, and how many coordinates each translation has. Going from the plane to space adds a coordinate to every translation and adds generators to most point groups. The classes multiply, and what each one carries multiplies too.

The polar orthorhombic class is the clearest single case. Its point group needs two generators, each translation has three coordinates, and each generator may carry a half in each of them — which is sixteen assignments before anything is identified, ten groups after. The plane’s largest class, 2mm on a rectangular lattice, has two generators with two coordinates each and comes to four assignments and three groups. Two generators in three dimensions instead of two, and four candidates become sixteen; that single change of coordinate count is most of the difference between the two dimensions.

The first step counts more things than a reader expects #

The class count is the one people underestimate, and the reason is that three different counts are all called classifications of the same thing.

In the plane there are five lattice types — oblique, rectangular, centred rectangular, square, hexagonal — which is a classification of lattices by their own symmetry. There are ten geometric classes, which is a classification of point groups by what they do to space. And there are thirteen arithmetic classes, which is a classification of point groups together with the lattice they act on, up to a change of integer basis. The last is the one the enumeration needs, because a group of matrices is not determined by the abstract group it is: the same mirror acting on a rectangular lattice and on a centred rectangular one is two different objects, and the two carry pm and pg on one hand and cm on the other.

Thirteen against ten is a small excess and it is entirely the mirror and 2mm appearing twice each, on the rectangular lattice and the centred one. In space the same excess is seventy-three against thirty-two, which is much larger, because there are fourteen lattice types for a point group to sit on rather than five and more of the point groups are compatible with several.

So the first step’s growth is the product of two growths already. The point groups multiply — ten, thirty-two, two hundred and twenty-seven — and so does the number of lattices each can sit on. That is why the class count multiplies by six to fourteen a dimension rather than by two or three, and it is why the first step is the one that looks like the hard one.

The third step subtracts, and it barely does #

The classification’s last step is a quotient rather than a multiplication, and its size is worth knowing because it is the step that decides what counts as the same group.

Two assignments of translations that differ by a relabelling of the axes describe one group, so the extension classes are divided by the action of the normaliser — the changes of basis carrying the point group to itself. In the plane that step removes exactly one: eighteen extension classes become seventeen groups, and the single merge is in the rectangular 2mm class, where a mirror along one axis with a glide along the other is the same group as the arrangement with the axes swapped.

One merge in thirteen classes is almost nothing, and the plane’s small numbers make it look like a detail. It is not, and the place it stops being a detail is one dimension up, where the same question produces two hundred and thirty or two hundred and nineteen depending on whether the changes of basis are allowed to reverse orientation. Eleven pairs survive the orientation-preserving quotient and merge under the full one — eleven merges where the plane has one — and in higher dimensions the merges are numerous and are where the arithmetic is delicate.

The third step is a subtraction and it is small beside the second’s multiplication. Nothing in the counts above would change much if it were dropped: eighteen instead of seventeen, and some number a little above 4,783 instead of 4,783. It matters for correctness and not for length, which is the opposite of the second step.

One step steps, and the other runs #

Dividing the two counts at every dimension separates them.

The class count goes 2, 13, 73, 710, 6,079, 85,311, multiplying by 6.5, 5.6, 9.7, 8.6 and 14.0. That is fast and it is steady: it is a search over finite groups of integer matrices, there are steadily more of them as the matrices get bigger, and the multiplier creeps up rather than taking off.

The groups-per-class ratio goes 1.00, 1.31, 3.15, 6.74, 36.5, 339, multiplying by 1.3, 2.4, 2.1, 5.4 and 9.3. The sequence is not monotone in its multiplier — the step from three dimensions to four is slightly gentler than the one before it — but the direction over six dimensions is unmistakable, and by the sixth the ratio is multiplying faster than the class count is.

So the explosion in the list is in the step that attaches translations. Finding the matrix groups is the step everyone thinks of as the hard one, because it is the one with a search in it; it is the one growing steadily. Attaching translations is a cohomology computation, which sounds like bookkeeping, and it is the one that runs away.

That has a consequence worth stating for anyone reading the counts. The ratio, not the class count, is what a further dimension mostly adds. Seven dimensions would have of the order of a million arithmetic classes and, if the ratio’s trend holds, several thousand groups apiece — which is why no seven-dimensional enumeration exists and why the sixth took the effort it did.

The bound explains the finiteness and not the length #

It is tempting to read Minkowski’s bound as the explanation of everything, since it is the first step of the chain and the only one with a formula.

The bound is a statement about one finite group of integer matrices: its order divides M(n). It says nothing about how many such groups there are up to conjugacy, and nothing at all about how many ways translations can be attached to one of them. Those are the two counts that actually grow.

In the plane and in space the bound happens to be larger than the answer, which encourages the wrong reading. By six dimensions M(6) is 2,903,040 and there are 28,927,922 groups — the answer has overtaken the bound, and there is no contradiction whatever, because the two numbers are counting different things. One is the largest a point group can be; the other is how many groups there are.

The same distinction runs through the argument these counts began with. Minkowski’s bound is about a whole group and the crystallographic restriction is about one operation; neither implies the other, and neither is a count. What makes the list finite and what makes it long are different facts, and keeping them apart is most of what this page is for.

Where the exactness stops #

Computed here. The plane in full: thirteen arithmetic classes, the number of extension classes each admits, and the seventeen groups those come to, by the cocycle computation the plane’s classification already needs. Six of the seventy-three classes of space, each enumerated from its generators by closing the group and identifying settings, and each checked against the number the International Tables record for it.

Everything above three dimensions is quoted. The counts 710 and 4,783 come from the machine enumeration of Brown, Bülow, Neubüser, Wondratschek and Zassenhaus in 1978; the five- and six-dimensional counts from later work in the same tradition. Nothing here reproduces any of them, and the table’s last column says so row by row.

Six classes are a sample and not a mean. They average 4.17 groups a class against the true 3.15, because they were chosen for being enumerable here rather than at random, and the classes with the largest point groups — which carry the most groups — are the ones hardest to enumerate. The sample supports the statement that space’s classes are not like the plane’s; it does not estimate the ratio.

And the ratio is an average over very unequal classes. In the plane it hides a distribution of ten ones, two twos and a three. In six dimensions an average of 339 groups a class certainly hides a much wider one, and nothing here says what shape it has.

The three steps are the algorithm’s steps, not the subject’s. Attributing growth to one of them is a statement about a particular way of counting — Zassenhaus’s, which is the only way anybody has counted past three dimensions. A different route to the same list would apportion the growth differently, and there is no sense in which the second step is where the groups “really” come from. What the ratio measures is where the work is when the list is produced this way, which is the question anybody producing one has to answer.

The first refusal is the sample being read as a mean, which is exactly the mistake the six classes invite. The second is a quoted count being taken for a derived one — the distinction the table’s last column exists to keep, and the one that would be quietly lost if the six rows were printed in the same style.

Who counted, and what it cost #

Fedorov and Schoenflies had the two hundred and thirty by 1891, by hand and by case analysis, and the two lists disagreed at first; reconciling them found errors in both. That is the largest enumeration anybody completed without a machine, and the ratio column says why: 73 classes at about three groups apiece is a few hundred objects, which a determined person can hold.

The four-dimensional count was produced in 1978 by Harold Brown, Rolf Bülow, Joachim Neubüser, Hans Wondratschek and Hans Zassenhaus, running Zassenhaus’s own algorithm of 1948. Seven hundred and ten classes at nearly seven groups apiece is a few thousand objects — past hand and comfortably inside a computer of that period. The five- and six-dimensional counts came later and the sixth is the last one anybody has completed.

The reason the sequence stops at six is the ratio rather than the classes. Enumerating eighty-five thousand arithmetic classes is a large computation and a routine one; computing the extensions of each, and then deciding which of them are the same group under the normaliser’s action, is where the twenty-nine million come from and where the work is.

Still open: what the distribution looks like #

The ratio is an average and every statement above is about an average. The distribution beneath it is reported nowhere here and is the obvious next question, because two very different shapes would produce the same mean.

The plane’s distribution is known here and is short: ten classes with one group, two with two, one with three. The natural guess is that the shape is roughly the same one stretched — a long tail of classes carrying one or two groups and a handful carrying very many — because the classes with large point groups have the most generators to attach translations to, and there are few of them. If that is right, the average of 339 in six dimensions is a statistic about a few enormous classes and describes almost none of them.

The alternative is that the tail thickens, so that a typical class in high dimension really does carry hundreds of groups. That would be a different fact about the subject: it would say that in high dimensions being non-symmorphic is the ordinary condition rather than the exceptional one, where in the plane four of the seventeen are non-symmorphic and in space a hundred and fifty-seven of two hundred and thirty.

The share of non-symmorphic groups is the quantity that would settle it, and it is computable one dimension at a time from data that already exists. It is computable here in the plane and for six classes in space and nowhere else, which is exactly the boundary every statement on this page runs into.