Thirteen ways to hold a lattice

The crystallographic restriction is a statement about one matrix. An integer matrix of finite order has an integer trace between −2 and 2, so 2 cos θ is an integer, so its order is 1, 2, 3, 4 or 6 and nothing else. One line, and it decides which rotations a lattice can have.

That answers a question nobody in crystallography actually asks. What a crystal has is not a rotation; it is a group of them, acting on one lattice at once. So the question the classification needs answered is how many such groups there are — and the answer is thirteen.

Thirteen is not a number this subject usually leads with, and it sits between two that it does. There are ten plane point groups and seventeen plane groups, and the gap at each end is the content.

What the thirteen are counting #

A finite group of symmetries of a lattice is a finite group of integer matrices — integer because the operations map lattice points to lattice points. Two such groups describe the same situation when a change of basis carries one to the other, and a change of basis is an integer matrix of determinant ±1. So the classification wanted is: the finite subgroups of GL(2,ℤ), up to conjugacy in GL(2,ℤ).

That is a computation and it is done here by search. Every finite subgroup fixes a positive definite form — average any form over the group — so it preserves some lattice metric, so it sits inside that metric’s holohedry, which has order at most twelve. And every such subgroup is conjugate into one of the two maximal holohedries: the square lattice’s group of order eight, and the hexagonal lattice’s of order twelve.

So the enumeration is finite and short: take every subgroup of those two groups, merge the ones that are conjugate, and count. Thirteen classes come out, with orders 1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 6, 8 and 12.

Ten geometric classes, and the three that split #

Collapse the thirteen by which abstract group each is, ignoring how it sits on the lattice, and ten remain: 1, 2, m, 2mm, 3, 3m, 4, 4mm, 6, 6mm. Those are the geometric crystal classes of the plane, and they are the ten a reader meets first.

Three of the ten split, and all three splits are about mirrors.

m splits into mp and mc. A mirror whose axis is a lattice vector holds a rectangular lattice; a mirror whose axis is a diagonal holds a centred one. The two are not conjugate in GL(2,ℤ) — there is no integer matrix of determinant ±1 carrying the first arrangement to the second — and that is exactly the distinction between pm and cm, which centring and why cm is not pm is about.

2mm splits the same way, into 2mmp and 2mmc, which is pmm against cmm.

3m splits into 3m1p and 31mp, which is this site’s oldest finding wearing arithmetic clothes. In p3m1 the mirrors run perpendicular to the lattice axes and in p31m along them, and the two are different groups for exactly the reason the two arithmetic classes are different: no integer basis change carries one family of mirror directions to the other.

The rotation-only classes never split. 1, 2, 3, 4 and 6 each hold their lattice in one way, because a rotation has no axis to align with anything. Splitting requires a reflection, and the split is always the same distinction: does the mirror lie along a lattice vector or along a diagonal?

Thirteen, from the other end #

Here is the check that makes the number mean something rather than being a number the search happened to produce.

A plane group is symmorphic when some choice of origin makes every operation’s translation part a lattice vector — when the group is a semidirect product of its lattice with its point group rather than a genuine extension. Testing that is a sweep over candidate origins and has nothing to do with matrices up to conjugacy.

Thirteen of the seventeen plane groups are symmorphic. The other four — pg, pmg, pgg and p4g — are not, and they are precisely the second occupants of the four classes that carry two groups.

So the number of arithmetic classes and the number of symmorphic plane groups are the same number, arrived at from opposite ends. One is a search over integer matrices with no wallpaper group anywhere in it; the other is a property of the seventeen groups’ own translations. They agree at thirteen, and the figure does not appear if they disagree.

That is a theorem rather than a coincidence: a symmorphic group is determined by its arithmetic class, since the class fixes the lattice and the point operations and the semidirect product then has no freedom left. One symmorphic group per class is the same statement in three dimensions, where the numbers are 73 arithmetic classes and 73 symmorphic space groups out of 230.

What each of the three counts is for #

The three numbers are not competing answers to one question. They answer three, and using the wrong one is a recognisable error.

Ten geometric classes answer what physical properties may a crystal have? Neumann’s principle works through the point group acting on space, and a tensor cannot tell a rectangular mirror from a diagonal one — what a trace decides is a statement about geometric classes, and there are ten of them in the plane and thirty-two in space.

Thirteen arithmetic classes answer what may the lattice and its point group be together? That is the question a diffraction pattern answers, since the reciprocal lattice carries both, and it is the question a structure determination has to settle before it can choose a space group.

Seventeen plane groups answer what may the whole pattern be? — the arithmetic classes plus the translations no origin removes.

Mixing the first two is the commonest confusion in this part of the subject, and it has a diagnostic: any statement of the form “the crystal class is cm” is confused, because cm is a plane group and its class is m holding a centred lattice.

The 2mm split, and what it does to a diffraction pattern #

The 2mm split is the one with the clearest experimental consequence, and it is worth following because it shows what an arithmetic class is in a laboratory.

Both classes have four operations: identity, half-turn, two mirrors. Both are the geometric class 2mm and no measurement of a physical property can tell them apart — the dielectric tensor of a crystal in one has exactly the shape it has in the other, since a tensor is blind to where the lattice points are.

A diffraction pattern is not blind to it. The centred lattice’s reciprocal lattice has systematic absences — half the reflections are missing, in the pattern centring and why cm is not pm sets out — and the primitive one’s does not. So the two arithmetic classes are distinguished by an experiment that cannot see the point group at all, and the point group is distinguished by an experiment that cannot see the lattice.

That is why the arithmetic class is the unit a structure determination works in. It is exactly what the diffraction pattern hands over: the lattice, from where the reflections are, and the point group, from how their intensities are related. What the pattern does not hand over is the translations inside the cell, which is the step from thirteen to seventeen and from 73 to 230.

What the search is, and what could go wrong with it #

Two subgroups are the same arithmetic class when some T in GL(2,ℤ) carries one onto the other, and T is found by sweeping integer matrices of determinant ±1 with small entries.

The failure mode is one-directional and silent. A sweep too small cannot merge two classes that a larger matrix would identify, so it reports too many classes, and each extra one looks like a discovery. Nothing about the output says the sweep was inadequate.

So the count is computed at more than one width and required not to move: thirteen with entries to ±2 and thirteen with entries to ±3. That is evidence rather than proof, and the proof is elsewhere — the classical result is that a conjugator can always be found with small entries once both subgroups are given in reduced bases, which is what the reduction machinery of the shortest basis supplies.

The second thing checked is the naming, because the labels are the part a search cannot supply. The two trigonal classes are named by whether their reflections fix a basis vector, and the check is that the class so named holds p3m1 and the other holds p31m — that way round. A naming rule that reads well and is inverted would produce a table nobody could catch by looking at it.

Why the two counts have to agree #

The coincidence of thirteen arithmetic classes and thirteen symmorphic groups is named as a theorem rather than an accident, and the theorem is one line in each direction.

From a class to a group. Given a finite group P of integer matrices acting on a lattice L, form the set of pairs — an operation of P together with a lattice translation — with the obvious composition. That is the split extension, it is a plane group, every one of its operations has a translation part in L, and so it is symmorphic by definition. One class, one group.

From a group to a class. Given a symmorphic plane group, choose the origin at which every translation part is a lattice vector. The linear parts then form a finite group of integer matrices acting on the lattice — an arithmetic class — and the group is recovered from it by the construction above. One group, one class.

The two constructions are inverse, so the counts are equal, and the equality holds in every dimension: seventy-three arithmetic classes and seventy-three symmorphic space groups, in three dimensions, for the same reason.

That is worth stating because it says what a symmorphic group is, rather than only what it is not. A symmorphic group is an arithmetic class with nothing added, and the non-symmorphic ones are the arithmetic classes with a translation part no origin removes — which is exactly the extension arithmetic the cocycle essay computes.

Where the thirteen are used #

The classes are presented here as an answer to a question, and they are also an input — the input to the only route this collection has to the seventeen that never draws anything.

The extension count begins with the arithmetic classes. For each of the thirteen, it enumerates the ways of attaching translations to the point group so that the composition closes, quotients by the origin shifts, and adds up. Nine of the thirteen admit only the trivial attachment and give one group each; four admit more; and the total is seventeen.

So the thirteen sit between the five lattices and the seventeen groups as a genuine intermediate stage, and the two steps are of different kinds. Getting from five to thirteen is a search over integer matrices — geometry, in the sense of which groups a lattice can hold. Getting from thirteen to seventeen is cohomology — arithmetic, in the sense of which translations can be attached.

That division is why the three counts answer three questions rather than refining one another. The lattice decides what the matrices can be; the matrices decide what the translations can be; and only the last step produces a plane group.

Where the exactness stops #

Everything above is exact. Integer matrices, integer conjugators, exact equality of sets of matrices. There is no tolerance anywhere and no measurement.

The classification is of groups acting on a lattice, not of crystals. Two crystals in the same arithmetic class can have nothing else in common; the class fixes what the symmetry can be and says nothing about what is at the lattice points.

And it is a plane classification. The three-dimensional analogue is larger and less tidy: 32 geometric classes, 73 arithmetic ones, 14 Bravais lattices, 230 space groups. The step from 32 to 73 is the same step as from 10 to 13 — the same point group holding differently centred lattices — and it happens far more often, because there are more centrings to choose from.

Who counted them, and when #

The arithmetic classes are Bravais’s and Frankenheim’s inheritance rather than anybody’s single discovery. Frankenheim enumerated lattice types in 1826 and got fifteen; Bravais corrected him to fourteen in 1848, and the correction was exactly an arithmetic-class argument — two of Frankenheim’s candidates turned out to be the same lattice in different bases.

The systematic version belongs to Fedorov and Schoenflies, working independently around 1891, whose enumerations of the 230 space groups pass through the arithmetic classes on the way. Zassenhaus gave the modern algorithm in 1948 — the one that turns a point group and a lattice into a finite calculation over cohomology — and the classification of finite subgroups of GL(n,ℤ) as a subject in its own right is due to the computational group theorists of the 1970s onwards, who have taken it as far as dimension six.

The numbers, dimension by dimension: 2, 13, 73, 710, 6,079, 85,308 arithmetic classes. The first is trivial, the second is this essay, the third is crystallography’s, and the rest are enumerations by machine.

The two maximal groups, and why the search is short #

The enumeration is short because of a fact worth stating on its own: there are exactly two maximal finite subgroups of GL(2,ℤ) up to conjugacy, of orders eight and twelve.

The reason is the reduction. A finite subgroup fixes a metric; a metric reduces to one of the five plane types; and the holohedries of those five have orders 2, 4, 4, 8 and 12, of which only the last two are not contained in a larger one. The oblique lattice’s group of order two sits inside every other; the rectangular and rhombic groups of order four sit inside the square’s eight and the hexagonal’s twelve respectively.

The merging is not a union, and the overlap is where the count would go wrong if it were. Four of the thirteen are conjugate into both maximal groups — the trivial group, the half-turn, the mirror on a centred rectangular lattice, and 2mm on the same — because those are exactly the classes whose matrices can be written on a square lattice and on a hexagonal one alike. Eight sit in the square’s holohedry, nine in the hexagonal’s, and eight and nine are seventeen, which is not the answer; taking the four shared ones once each is what makes it thirteen. That is a small thing and it is the kind of small thing an enumeration by hand gets wrong.

So the whole classification is the subgroup lattices of two small groups, merged. That is why the answer can be stated with confidence rather than as the output of a long computation, and why the same argument in three dimensions is a great deal harder: there the maximal finite subgroups of GL(3,ℤ) number four rather than two, and in higher dimensions the count of maximal groups is itself a research problem.

The word “class” carries three meanings, and they are worth separating #

The vocabulary here is a standing trap, and the three senses are all in use in the same paragraphs of the same textbooks.

A geometric crystal class is a point group up to conjugacy in the orthogonal group — ten in the plane, thirty-two in space. It is what a physical property sees.

An arithmetic crystal class is a point group together with its lattice, up to conjugacy in the integer general linear group — thirteen in the plane, seventy-three in space. It is what a diffraction pattern sees.

A Bravais class is a lattice up to the same equivalence, classified by its own holohedry — five in the plane, fourteen in space. It is what a reduced cell sees.

The three refine each other in one direction: every arithmetic class has a geometric class and a Bravais class, and neither of those determines it. The commonest confusion is between the first two, and it has a symptom worth remembering — a statement that names a lattice type and a point group in the same breath is an arithmetic-class statement, whatever it calls itself.

Where the ladder goes next #

This rung takes the restriction from one matrix to a group of them. Two rungs sit above.

The extension problem. Given an arithmetic class, how many plane groups sit over it? The answer is a cohomology computation, and in the plane it produces the four non-symmorphic groups and nothing else. A space group is an extension does this in three dimensions, where the same calculation produces 157 non-symmorphic groups over 73 classes.

Higher dimensions. The finite subgroups of GL(n,ℤ) are finite in number for every n — a theorem of Jordan’s — and Bieberbach’s theorems turn that into the finiteness of the crystallographic groups in every dimension. Where five-fold symmetry becomes legal is the four-dimensional case, and that essay asks it about a single rotation. Asking it about a whole group is this essay one dimension up, and the answer is 710 rather than 13.