Thirty-two, and no others

A crystal’s symmetry has two halves. One of them is the lattice — the repetition that goes on forever, and which was settled in three dimensions by the fourteen Bravais lattices. The other is what happens at a single point: the rotations, reflections and inversions that leave that point where it is and carry the crystal onto itself.

That second half is a finite group, and there are exactly thirty-two of them.

The number is old — Hessel had it in 1830 and nobody noticed, Gadolin had it again in 1867 and it stuck — and it is usually met as a list to be accepted. It does not have to be. The whole classification is a finite search over finite objects, and this essay runs it.

What is being counted #

A crystallographic point group is a finite group of orthogonal transformations fixing a point, which is also a symmetry group of some lattice. The second clause is the entire content: without it the finite groups fixing a point are infinite in number, since a rotation through any angle at all generates one.

With it, the crystallographic restriction applies. Every operation must be an integer matrix in the lattice’s own basis, so its trace is an integer, so the only rotation orders available are 1, 2, 3, 4 and 6 — and their rotoinversions. Ten kinds of operation, and no others.

Two point groups count as the same crystal class when one is carried onto the other by a change of basis. That is the right equivalence and it is easy to state loosely and get wrong: a two-fold axis along a cube edge and a two-fold axis along a face diagonal are two different subgroups of the cubic holohedry, occupying different positions in the lattice, and they are one crystal class, because a rotation of the whole picture takes one to the other.

Why two groups are enough to search #

The search would be hopeless if it had to range over all finite subgroups of the orthogonal group. It does not, because of a containment that costs nothing to state and everything to the method:

Every crystallographic point group is a subgroup of m3̅m or of 6/mmm.

Those are the holohedries of the cubic and hexagonal systems — the full symmetry groups of a cubic lattice and of a hexagonal one, of order 48 and 24. Every other holohedry is itself a subgroup of one of them: 4/mmm, mmm, 2/m and 1̅ all sit inside m3̅m, and so does 3̅m, along a body diagonal. And a point group is by definition a subgroup of the holohedry of some lattice it preserves.

So the search is over the subgroups of two finite groups, one of order 48 and one of order 24. That is small enough to do exhaustively, which is the difference between a classification and a claim.

Finding every subgroup #

Testing every subset of a group of order 48 for closure is 2⁴⁸ tests. The method that works instead is cyclic extension, and it is three lines:

Start with the trivial subgroup. Take every subgroup found so far, adjoin one element it does not contain, and close. Repeat until nothing new appears.

It terminates because each adjunction at least doubles the order, so no chain is longer than log₂ 48. It is complete because every subgroup K is generated by its own elements: adjoining them one at a time from the trivial group is a chain of subgroups inside K that reaches K, and the search walks every such chain.

That gives 98 subgroups of m3̅m and 54 of 6/mmm. Both counts are asserted. The first is a number that appears in the literature on the subgroup structure of the octahedral group and agreeing with it is a check on the method rather than a use of it.

The first quotient: same group, different orientation #

98 and 54 are not classifications, because most of those subgroups are the same group sitting in the lattice more than one way.

The cubic holohedry contains two-fold axes along the cube edges and two-fold axes along the face diagonals. Each generates a subgroup of order 2, and there are nine such subgroups — three edges and six diagonals — but there are only two conjugacy classes of them, because no operation of the cube carries an edge to a face diagonal. Both classes are nevertheless the class 2.

Quotienting by conjugacy inside the holohedry takes 98 to 33 and 54 to 32. That is the largest single step of the reduction and it is entirely mechanical: conjugate each subgroup by every element of the holohedry, collect the orbit, and count the orbits.

It also leaves something behind, and the leftover is the reason the enumeration is not finished at this point. Conjugacy inside a holohedry is a finer relation than conjugacy in space. Two subgroups can fail to be related by any symmetry of a cubic lattice and still be related by a rotation — a rotation that is simply not a symmetry of that lattice.

The second quotient: merging the two lists #

Thirty-three plus thirty-two is sixty-five, and the answer is thirty-two, so a great deal is about to be identified. Thirteen classes appear in both holohedries: everything with no three-fold or four-fold or six-fold axis lives happily in either, and so do the trigonal classes, which sit in a cubic lattice along its body diagonal and in a hexagonal one along c.

The merge is proposed by an element signature — the count of operations of each of the ten crystallographic types. Two subgroups with the same census of operations are candidates for being the same class.

The signature is a fingerprint, and a fingerprint that returns the right answer is still a fingerprint. It gives thirty-two. That is correct, and being correct is not the same as being a complete invariant — the argument for checking it, and what checking it found, is the next rung of this ladder and is worth its own essay. What matters here is that the merge is not taken on the fingerprint’s word: for every pair merged, a rational matrix carrying one group onto the other is constructed, and the enumeration refuses to return if any pair fails to produce one.

Thirty-three classes and thirty-two, minus thirteen counted twice, is thirty-two: twelve that exist only in a cubic setting, seven only in a hexagonal one, thirteen in both.

How the thirty-two distribute #

The classes do not spread evenly over the seven crystal systems, and the shape of the distribution says something about the subject.

The counts per system are two, three, three, seven, five, seven and five — triclinic smallest, tetragonal and hexagonal largest — and a system’s classes are exactly the subgroups of its holohedry that are not subgroups of any smaller one, which is what makes the assignment a computation rather than a convention.

A class belongs to the system whose holohedry is the smallest one containing it, and “smallest” is doing real work in that sentence. Class 2 is a subgroup of m3̅m, of 6/mmm and of 4/mmm as well as of 2/m; its system is monoclinic because 2/m is the smallest of them. Reading the assignment off the holohedry a class happened to be found in would put half the classes in the cubic system.

The cubic system is also the one identified by a count rather than by a maximum. Every other system is recognised by its highest-order axis — a six-fold means hexagonal, a four-fold with nothing above it means tetragonal. Cubic has no axis higher than four, and what marks it is having four three-fold axes rather than one: eight operations of order three, along the four body diagonals. A classifier that asked only for the highest order would file the cubic classes as tetragonal or trigonal, which is the second place in this enumeration where the obvious test is the wrong one.

The names come out of the operations #

A classification that produces thirty-two anonymous groups is only half of one. The Hermann–Mauguin symbol of each is derived here rather than attached: each crystal system has an ordered list of symmetry directions, and the symbol reports, for each direction in turn, the highest-order axis lying along it and whether a mirror is perpendicular to it.

Take 4/mmm as the worked case. The three tetragonal directions are c, then ⟨100⟩, then ⟨110⟩; the first carries a four-fold with a mirror across it and so do the other two with two-folds, so the full symbol is 4/m 2/m 2/m and the short one is 4/mmm. Nothing in that reading is looked up, and the derivation does the same for the other thirty-one — with one genuine convention in it and one exception, and the exception is orthorhombic.

The derivation is required to reproduce all thirty-two of the Tables’ symbols, and it does. That agreement is the strongest single check in the whole enumeration, because the symbols encode where each element sits and not merely how many there are — a classification that had merged two classes or split one would produce a symbol nobody recognises, or two identical ones.

This is the same arrangement the notation index uses for the seventeen plane groups, one dimension up. The rule there was that reading the hexagonal direction families in the wrong order must derive p3m1 as p31m, and the gate carries the refusal. The rule here is the same shape.

The diagram is generated, and is checked by counting #

Every stereogram on this page is the orbit of one general direction under one group. That is the same construction the site’s plane-pattern figures use — the orbit is the pattern — and it carries the same failure mode.

A direction that is not general produces a picture of a smaller group. If the chosen direction happens to lie on a mirror plane of the group being drawn, the mirror maps it to itself, two operations produce one pole, and the diagram comes out with half as many marks as the group has operations. It is a perfectly ordinary stereogram. It is the stereogram of a different class, printed under this one’s name, and nothing about it looks wrong.

This is the oldest finding on this site arriving for the third time. In the plane it was that a single dot verifies only eleven of the seventeen groups, because the motif must be a comma and a dot is too symmetric. In space it was that the orbit of a single point under P2₁ acquires an inversion centre. Here it is a direction rather than a motif, and the remedy is identical: count what came out and compare it with what went in. Every diagram asserts that its pole count equals its group’s order, and the direction used is one chosen to be general in all thirty-two rather than in whichever class is being drawn.

Where the exactness stops #

Three limits, and they are limits of what has been claimed rather than of the arithmetic.

The classification is of point groups, not of crystals. A crystal belongs to a class; a class does not describe a crystal. Two materials in class mmm can be as different as it is possible for two solids to be.

Thirty-two is a count of classes, not of groups. There are 98 subgroups of m3̅m alone. The reduction from 65 conjugacy classes to 32 crystal classes is a decision about what “the same” means, made twice, and both decisions are stated above rather than buried.

Nothing here decides the 230. Getting from the thirty-two point groups and the fourteen lattices to the two hundred and thirty space groups requires the translations, and that is a different and much longer computation. This site enumerates the arithmetic classes it can and says so; 230 remains a number from the literature in every essay that uses it.

What was verified, and how #

The enumeration is a function that either returns thirty-two classes or throws. Five counts are asserted along the way — 98, 33, 54, 32, and the total — and they are asserted separately on purpose, because a search that missed two subgroups and a quotient that failed to merge two classes would give the right answer for reasons that were both wrong.

Beyond those, three properties are required to hold before the list is handed to any figure:

• every merge carries an explicit conjugating matrix, so no class rests on a fingerprint alone;
• all seven holohedries appear among the thirty-two, which is the containment the whole method assumes and would otherwise never test;
• the thirty-two derived symbols are thirty-two distinct symbols, and are the ones the Tables record.

The third is where a real error would surface first. A classification with a class missing has thirty-one symbols; one with a class split has two derivations landing on the same name.

Thirty-two classes, and rather fewer groups #

The thirty-two are counted up to a change of basis, which is a geometric equivalence. Counted up to abstract isomorphism — ignoring how a group sits in space and asking only what its multiplication table is — the number falls, and the drop is instructive.

Three of the classes are 222, mm2 and 2/m. All three have four operations, all three are abelian, and every non-identity element of each squares to the identity: as abstract groups they are the same group, the one with two independent generators of order two. What distinguishes them is entirely how they act — three perpendicular two-folds, one two-fold with two mirrors through it, one two-fold with a mirror across it — and none of that is visible in a multiplication table.

Running the identification over all thirty-two leaves eighteen abstract groups. So more than a third of the classification is recording positions rather than structures, which is exactly the property the Hermann–Mauguin symbol is built to carry and the Schoenflies symbol largely is not.

It is also why an enumeration by abstract group would produce the wrong answer twice over. It would merge classes that a crystal distinguishes — 222 and mm2 behave completely differently, permitting different properties and forbidding different ones — and it would offer no way of writing down which one a given crystal has.

Geometric classes, and the arithmetic ones underneath them #

There is a second refinement of the same count, going the other way, and it is the one the space-group enumeration needs.

A geometric crystal class is what this page counts: a point group up to a change of basis by any invertible matrix. An arithmetic crystal class is finer — a point group together with the lattice it acts on, up to a change of basis by an integer matrix. Two crystals whose point groups are the same and whose lattices are differently centred belong to one geometric class and two arithmetic ones.

The plane has ten geometric classes and thirteen arithmetic ones, the extra three coming from the rectangular and centred-rectangular lattices carrying the same point groups. Space has thirty-two geometric and seventy-three arithmetic.

The distinction matters because the arithmetic classes are the input to everything that comes next. Building the space groups means attaching translations to a point group acting on a particular lattice, and a point group acting on a primitive lattice and the same point group acting on a centred one admit different sets of translations. So the enumeration on this page is a step towards the two hundred and thirty and not the first half of it: the count that feeds forward is seventy-three, and thirty-two is what seventy-three becomes when the lattice is forgotten.

Where this goes #

Thirty-two is the beginning of the field rather than the end of it. The classes matter because they decide things: which of them can hold a molecule of one handedness, which of them a diffraction experiment can tell apart, and — the part that reaches outside crystallography entirely — which physical properties a crystal in each class is permitted to have at all, before anybody measures one.

That last is Neumann’s principle, and it turns each of these thirty-two diagrams into a statement about elasticity, piezoelectricity and optical rotation. The eleven that contain the inversion turn out to be the eleven a diffraction pattern reports, which is why a structure determination begins with an eleven-way answer rather than a thirty-two-way one.

Both refinements are counts of the same objects under different equivalences, and neither is more correct than the other. Eighteen is the answer to how many multiplication tables; thirty-two to how many ways of acting on space; seventy-three to how many ways of acting on a lattice. A statement quoting one of them for a question about another is the standard way this part of the subject goes wrong, and it is why each of the three carries its adjective.