Why there is a list at all

Every essay on this site that counts something has taken for granted that the counting stops.

Five plane lattices. Seventeen wallpaper groups. Seven friezes. Thirteen arithmetic classes. Thirty-two crystal classes. Fourteen Bravais lattices, two hundred and thirty space groups, eleven Laue classes, five parallelohedra. Each of those numbers is produced here by an enumeration rather than quoted, which the site is strict about — and every one of those enumerations is a search that terminates, over a space that is not obviously finite.

The space really is not obviously finite. A symmetry of a lattice is an integer matrix of determinant ±1, and there are infinitely many of those; a group of them could in principle be arbitrarily large. Nothing about integrality by itself stops a lattice having a symmetry group of order ten thousand.

The question this essay is about is the one none of the others asked. Why is there a list?

The three things that must be finite #

The classification of plane groups is finite because three counts in a row are finite, and it is worth separating them because they fail for different reasons.

Finitely many lattice types. A lattice’s own symmetry group — its holohedry — is a finite group of integer matrices preserving a positive-definite form, and lattices are sorted by which group that is. Five in the plane, fourteen in space once centring is counted.

Finitely many finite groups of integer matrices. This is the hard one, and it is what the rest of this essay is about. Sorted up to a change of integer basis they are the thirteen arithmetic classes in the plane and seventy-three in space.

Finitely many extensions of each. Given a lattice and a point group acting on it, the ways of attaching translations to the generators is a finite calculation — the argument that turns thirteen into seventeen — because the translation attached to each generator is determined modulo the lattice and the possibilities form a finite group.

Take away the second and the whole edifice goes. That is the one Minkowski settled, and he settled it twice over: once with a bound, and once with an argument that produces no bound at all and is the more surprising of the two.

The bound #

For each dimension there is an integer M(n) that the order of any finite group of rational matrices must divide:

M(n) is the product over primes p of p raised to the sum of ⌊n / p^(k−1)(p − 1)⌋ over k = 1, 2, 3, …

Both the sum and the product are finite for a reason a reader can check by looking. The k-th term dies as soon as its denominator passes n, so the sum has at most about log₂ n terms; and the whole exponent is zero once p − 1 exceeds n, so the product runs over the primes up to n + 1.

A divisibility is much stronger than an upper bound. M(2) = 24 does not merely say that no plane lattice has more than twenty-four symmetries; it says that the number of symmetries divides twenty-four. So orders 5, 7, 9, 10 and 11 are excluded outright, before any geometry is done, and what remains to be checked is a short list.

Twenty-four is not attained; forty-eight is #

In the plane the bound is not reached. The largest finite group of integer matrices in two dimensions is the hexagonal holohedry, of order twelve — six rotations and six reflections — and twenty-four is twice that. There is no plane lattice with twenty-four symmetries and there never could be; the bound is honest about what it is, which is a divisibility argument that does not know any geometry.

In three dimensions it is reached exactly. M(3) = 48, and the cubic holohedry m3̅m has forty-eight operations: twenty-four rotations and their products with inversion. This site derives that group by enumerating the integer matrices preserving a cubic metric, which is the same derivation the fourteen Bravais lattices come from, so the agreement is between a number-theoretic formula and a metric computation with nothing in common.

Above three dimensions the gap widens. M(4) = 5,760 against a largest group of 1,152; M(6) is nearly three million against a hundred thousand. The bound remains true and stops being informative, which is the usual fate of a bound proved without reference to the objects it bounds.

The bound is about groups; the restriction is about operations #

There is a trap here, and it is the reason this essay sits in the restriction field rather than beside the classification.

The crystallographic restriction says that a lattice in n dimensions admits an operation of order k exactly when Euler’s totient of k is at most n — which gives orders 1, 2, 3, 4 and 6 in the plane, and the same five in space. That is a statement about one operation.

Minkowski’s bound is a statement about a whole group. The two are independent, and the clearest way to see it is to find an order that one permits and the other forbids.

Eight divides twenty-four, and there is no eight-fold rotation in any plane lattice. Twelve divides twenty-four, and no twelve-fold rotation exists either — though a group of order twelve does, which is exactly the point: the hexagonal holohedry has order twelve and its largest rotation is six-fold. Order twelve as a group and order twelve as an operation are different facts and only the first occurs.

The seventeen themselves make the point again. Their point groups have orders 1, 2, 3, 4, 6, 8 and 12 — seven values, every one a divisor of twenty-four, and the only divisor of twenty-four that never occurs is twenty-four itself. The bound is therefore not slack by much on this list: of the twenty-three numbers below it, sixteen are not divisors and are excluded before any geometry is done, and every divisor but the last is realised by an actual plane group.

That closeness is a coincidence rather than a theorem, and it is worth naming as one, because it does not survive going up. In three dimensions the divisors of forty-eight are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48, and all ten occur among the crystal classes and holohedries. In four dimensions the bound has 5,760 divisors and only a handful are ever realised. The bound and the reality happen to agree closely in exactly the two dimensions somebody was going to check by hand.

What a search has to know before it can stop #

The abstract statement has a very concrete consequence, and it is visible in how the enumerations on this site are actually written.

The arithmetic classes are found by taking the two maximal holohedries — square and hexagonal — listing every subgroup of each, and merging the results under conjugacy by integer matrices. That search is finite because the holohedries are finite, and it is complete only because every finite group of integer matrices is conjugate into one of them. Both halves are needed, and the second is the one that needs the theory.

The fourteen Bravais lattices are found the same way one dimension up: twenty-five candidate cells, reduced to fourteen by finding which pairs describe the same lattice on different axes. The search over candidates terminates because the list of holohedries does.

The thirty-two crystal classes are enumerated by closing up sets of generators and merging by conjugacy, with the same guarantee underneath.

And where the guarantee is missing, the site says so. Counting the sublattices of a given index is finite for a different reason — there are finitely many subgroups of index n in ℤ² for elementary reasons, and the count is a divisor sum. The higher-dimensional restriction is decided by cyclotomic degrees rather than by any bound of this kind. Each enumeration has its own stopping argument, and the one in this essay is the one that covers the largest number of them.

The practical form of the theorem is therefore not “the answer is at most twenty-four” but “a search over the subgroups of the maximal holohedries is exhaustive”, and that is the sentence every enumeration here depends on.

Bieberbach, and the theorem underneath the theorem #

One assumption has been carried silently through all of the above: that a crystallographic group has a lattice — that its translations form a subgroup of full rank and finite index, so that the point group is finite in the first place.

In two and three dimensions that can be taken as the definition, and this site does take it so: a plane group is defined here as a lattice with operations attached. In general it is a theorem, and it is the first of Bieberbach’s, proved in 1911: a discrete group of isometries of n-dimensional space with compact quotient contains n independent translations, and its point group is finite. Without it, “the point group” is not obviously a finite object and the bound has nothing to bound.

The second — isomorphic crystallographic groups of the same dimension are conjugate in the affine group — is what makes a classification of patterns into a classification of groups, and it is what the abelianisation essay leans on when it says the seventeen are seventeen distinct abstract groups.

The third is the answer to Hilbert: finitely many in each dimension, up to affine conjugacy.

The three together are why a phrase like “the seventeen wallpaper groups” is well formed. Each of them removes an ambiguity that would otherwise make the count depend on what exactly was being counted — patterns, groups, or descriptions — and the site’s habit of insisting on which of the three a number refers to is inherited directly from them.

Where the exactness stops #

The formula is exact and the arithmetic is integral. Floors of integer quotients, powers of primes, and a product. There is nothing to round.

The divisibility is a theorem, and what is computed here is the check. Every one of this site’s thirteen arithmetic classes has an order dividing twenty-four; every three-dimensional holohedry has an order dividing forty-eight. That is thirteen checks and seven, against a statement about infinitely many groups. Passing them is consistency, not proof, and the proof is Minkowski’s.

Two of the attained maxima are computed here and four are quoted. In one, two and three dimensions the largest finite group of integer matrices is the largest holohedry, which this site derives from metrics. The values for four, five and six dimensions are results in the literature; nothing here enumerates the finite subgroups of GL(4,ℤ), and the table marks which column is which.

Finiteness of orders is not finiteness of groups. Knowing that every finite subgroup of GL(2,ℤ) has order dividing twenty-four does not by itself say there are thirteen up to conjugacy — it says there are finitely many orders, and one still needs that each order admits finitely many classes. That second step is a separate argument, and the enumeration this site runs is a search over subgroups of the two maximal holohedries, which is only sufficient because every finite group of integer matrices is conjugate into one of them.

Who found it, and when #

Hermann Minkowski proved the divisibility in 1887, in a paper on the arithmetic of quadratic forms. He was after the reduction theory of forms, not crystallography, and the result about matrix groups came out as a tool. It is one of a small number of theorems that a crystallographer relies on daily without ever meeting: it is the reason the International Tables are a book of finite length.

Camille Jordan had shown in 1878 that a finite subgroup of GL(n,ℂ) has an abelian normal subgroup of bounded index — a much more general finiteness, with a much worse bound — and the special case of integer matrices is where the answer becomes an explicit product over primes.

David Hilbert’s eighteenth problem, in 1900, asked whether there are finitely many essentially different crystallographic groups in each dimension. That the answer is yes for two and three dimensions was known from the enumerations of Fedorov, Schoenflies and Barlow in the 1890s; that it is yes in every dimension is Bieberbach’s, in 1911 and 1912, and it is the theorem that makes the question a good one rather than a lucky one. Bieberbach’s argument gives no bound on how many, and the counts are known only up to dimension six: 17, 219, 4,783, 222,018, 28,927,915.

The counting the other way round is worth noticing. The number of groups grows explosively while the bound on any one group’s point symmetry grows slowly, and the reason is that almost all of the growth is in the extensions rather than in the point groups. Higher dimensions do not have wildly more symmetric lattices; they have wildly more ways of attaching translations.

Where this ladder goes #

The bound answers how large, and it does it with a formula that a reader can evaluate in a minute. It does not answer why finite — the formula presupposes the theorem it comes from.

The next rung is the other half of Minkowski’s answer, and it is the cheaper and stranger one. Reduce every matrix modulo three. Any finite group of integer matrices is carried faithfully into a finite group of matrices over the integers modulo three, which is a group with 48 elements in the plane and 11,232 in space, and finiteness follows immediately with no bound computed and no geometry used.

The same argument modulo two is false, and it fails on the most familiar matrix in crystallography — which makes the next rung a rare thing in this subject: a proof whose hypothesis can be shown to be necessary by producing the counterexample, rather than merely stated and taken on trust. Minus the identity is a symmetry of every lattice there is, and it is congruent to the identity modulo two. Whatever else the argument does, it must not be allowed to prove that an inversion centre has infinite order.

The theorem that fills the last gap #

The essay ends by naming what the bound does not give: finitely many orders is not finitely many groups. The statement that closes it has a name, and it is worth having because it is the one the whole classification actually rests on.

Jordan–Zassenhaus. For each dimension there are only finitely many conjugacy classes of finite subgroups of the integer matrices. Not finitely many orders — finitely many groups, up to a change of basis, which is exactly the equivalence a classification uses.

Both halves of the name are doing work. Jordan’s contribution, from the 1870s, bounds the order of a finite subgroup of the rational matrices in terms of the dimension. Zassenhaus’s, much later, is the step from bounded order to finitely many classes — a statement about integral representations rather than about orders, and the harder half.

With it, every count on this site is a count of something finite by a theorem rather than by a search that stopped. Thirteen arithmetic classes in the plane, seventy-three in space, and 710 in four dimensions: each of those numbers is the answer to a question guaranteed in advance to have one.

Without it the searches would still terminate and would prove nothing. A sweep over integer matrices with entries up to some bound finds a list, and no amount of widening the bound establishes that the list is complete — that is the one-directional failure this collection’s own enumerations warn about. The theorem is what converts the sweep from evidence into a proof, and it does so before any sweeping happens.

How the numbers were actually got #

The theorem says the lists are finite and does not produce them, and the gap between those two is where the modern computations sit.

The three-dimensional lists were produced by hand. Fedorov and Schoenflies enumerated the two hundred and thirty in 1891 by case analysis, working through the arithmetic classes and their extensions, and the errors each of them made were found by comparing the two lists.

The four-dimensional lists were produced by machine, in the 1970s, by an algorithm following the same three steps this essay separates: find the lattice types, find the finite groups of integer matrices acting on each, and enumerate the extensions. Its answer — 4,783 space groups in four dimensions, from 710 arithmetic classes and 64 Bravais types — is not checkable by hand at any stage.

That is why the theorem matters more there than here. In two and three dimensions the lists are short enough that two independent enumerations can be compared, and the comparison caught real errors. In four they cannot be, and the guarantee that the computation was looking for a finite answer is a substantial part of the reason to believe it.