Eleven, eleven and ten

Every finite group of rotations of space is cyclic, dihedral, or one of three exceptions, and the list comes out of counting the poles the rotations fix. The five regular solids come out of the same count. Both are lists of rotations — motions a solid object can perform on itself without being taken apart — and the crystallographic restriction cuts the rotation groups down to eleven that a lattice can hold.

The crystal classes are mostly not rotations. There are thirty-two of them, found by taking every subgroup of the two largest holohedries and merging whatever a change of basis identifies, and twenty-one of them contain at least one operation no solid object can perform: a mirror, a centre of inversion, a rotoinversion. The search that finds them is exhaustive and says nothing about where they come from. It returns the twenty-one with improper operations the same way it returns the eleven without, as entries in a list.

They come from nowhere new. Every one of the twenty-one is one of the eleven rotation groups, read in one of two ways, and the reading is short enough to be a theorem rather than a search.

The determinant cuts every group in two #

Every operation of a point group is an orthogonal matrix, and its determinant is +1 or −1: plus for a rotation, minus for anything that reverses handedness. The determinant of a product is the product of the determinants, so the determinant is a homomorphism from the group onto a group of order at most two, and the operations it sends to +1 — the rotations — form a subgroup. Call it K. It is either the whole group or exactly half of it, because the kernel of a map onto a group of order two has index one or two.

That leaves three possibilities, and nothing else can happen.

The group is all rotations. K is everything and the group is proper. Eleven classes are like this, and they are the eleven a crystal of a single enantiomer may have.

The group contains the inversion. The inversion is the matrix $-I$, which sends every vector to its negative. It commutes with every matrix there is, so if it belongs to the group then every improper operation is $-I$ times a rotation, and that rotation lies in K. The group is K with the inversion adjoined — a direct product of the rotations with a group of order two — and nothing about it is new except the centre.

The group is improper and has no centre. This is the case with content in it. Take each operation M and replace it by $\det(M)\,M$: leave the rotations alone and negate everything else. A negated improper operation has determinant +1, so the result is a set of rotations. And the replacement preserves products, because $\det(AB)\,AB$ is $\det(A)\,A$ times $\det(B)\,B$ — determinants multiply, and a sign is a scalar that commutes with any matrix. So the set of rotations is itself a group, it has as many elements as the original, and the replacement is an isomorphism between them.

Call that rotation group G′. It contains K untouched, and K sits inside it at index two. Reading the construction backwards, the original group is G′ with the half of G′ outside K negated — a proper group, twisted. Negating the identity would give the centre, and the identity is always in K, so a twist can never produce the centre and the third case never overlaps the second.

The converse holds as well. Every rotation group with a subgroup of index two can be twisted this way, and the result is always a group: a subgroup of index two is normal, its complement is a single coset, and negating a whole coset is exactly what the replacement does in reverse. So an improper point group is one of exactly two things — a rotation group with the centre added, or a rotation group together with a chosen half of it — and classifying the improper groups is the same job as classifying halves.

Eleven with the centre #

The centred case is the easy one, and it is already familiar under another name. Adjoining the inversion to each of the eleven proper classes gives eleven groups of twice the order, and no two coincide, because the rotations of each recover the proper class it was built from. They are 1̅, 2/m, mmm, 4/m, 4/mmm, 3̅, 3̅m, 6/m, 6/mmm, m3̅ and m3̅m — which are the eleven Laue classes, the classes the symmetry of a diffraction pattern can tell apart.

That is not a coincidence either. Without anomalous scattering, Friedel’s law adds a centre to whatever symmetry a crystal has, so what a pattern shows is the class with the inversion adjoined, and by the theorem that is the class’s rotations with the inversion adjoined. The symmetry of a diffraction pattern reads a crystal’s rotations and nothing else. Two crystals whose classes share their rotations — 4, 4̅ and 4/m, or 432, 4̅3m and m3̅m — give patterns of identical symmetry, and the list of Laue classes is the list of proper classes wearing a centre.

The centred column also settles something before it can become a puzzle. The class 3̅ contains the centre, because a three-fold rotoinversion performed three times is the inversion itself, and the same is true of a rotoinversion of any odd order. That is why 3̅ and 3̅m sit in the middle column and not the last, and why no class could be a three-fold rotation twisted.

Ten with half of themselves negated #

The twisted case needs a subgroup of index two, and not every rotation group has one.

Fourteen halves exist among the eleven proper classes, and they produce ten classes. The difference is repetition of a familiar kind. 222 has three two-fold axes and any one of them can be the half that is kept, but the three results are a single class, mm2, with its polar axis along a, along b or along c. 422 has two subgroups of the shape of 222 — one with its two-fold axes along the cell edges and one with them along the diagonals — and both twist to 4̅2m, once in the orientation written 4̅2m and once in the one the Tables write 4̅m2. 622 does the same with its two copies of 32. In each case the halves are carried onto each other by a motion outside the group — a permutation of the axes for 222, a turn of forty-five degrees for 422, a turn of thirty for 622 — so they are one class drawn in different orientations.

Three proper classes have no half at all. 1 and 3 have odd order, and a group of odd order has no subgroup of half its size. The third is 23, the rotations of a tetrahedron, which has order twelve and no subgroup of order six. It is the standard first counterexample to the converse of Lagrange’s theorem, and it has turned up once already as a fact about crystals: a molecule at a site of symmetry 23 cannot be disordered over two orientations, for the same missing subgroup. Here the gap means there is no class built by twisting 23. The only improper cubic class without a centre, 4̅3m, is 432 twisted with 23 as the half left alone — the tetrahedral rotations appear in it as what is kept, never as what is twisted.

The stereograms make the construction visible in a way the matrices do not. A pole is the image of one general direction under one operation, and negating the operation sends its pole straight through the centre of the sphere — to the opposite side of the drawing, with a filled mark becoming an open one or the reverse. A twist is therefore a drawing in which exactly half the marks have crossed the page and half have stayed, and which half stayed is the choice of K.

Keeping the four-fold rotations of 422 leaves their four poles in place and sends the four belonging to the in-plane two-fold axes across, and those two-fold axes become the mirrors that contain the four-fold axis of 4mm. Keeping 222 instead sends the quarter-turns across, and the four-fold axis becomes a 4̅. The two drawings of 4̅2m differ only in whether the kept two-folds lie along the edges or the diagonals, which is a statement about axes and not about the group.

That completes the count. Eleven proper classes, eleven centred and ten twisted make thirty-two, and the improper half of the classification has been reduced to listing the subgroups of index two in eleven small groups.

Why a 4̅ can lack a centre and a 3̅ cannot #

The twist negates a particular set of operations — the ones outside K — and which operations those can be is decided by a one-line argument about orders.

An operation of odd order is a square. If an operation has order n and n is odd, it is the square of its own (n + 1)/2-th power. And every square lies in every subgroup of index two, because dividing by such a subgroup leaves a group of order two, in which everything squares to the identity. So the identity and every three-fold rotation always lie in the half that is kept, and no twist ever negates them.

That argument is the whole of the table. Negating a half-turn gives a mirror, since the half-turn fixes an axis and reverses the plane perpendicular to it, and the negative of that fixes the plane and reverses the axis. Negating a four-fold rotation gives a 4̅ and negating a six-fold one gives a 6̅. Negating a three-fold rotation would give a 3̅ and negating the identity would give the centre, and neither ever happens, so in all ten twisted classes those two columns stay blank. A twisted class is its partner with some half-turns, quarter-turns and sixth-turns turned into mirrors, 4̅ and 6̅, and with nothing else changed.

The same arithmetic explains a pattern the symbols show without explaining. 4̅ and 6̅ occur in classes with no centre — 4̅, 4̅2m, 4̅3m, 6̅ and 6̅2m — while 3̅ never does. A rotoinversion of even order never reaches the inversion among its powers; a rotoinversion of odd order always does. A Hermann–Mauguin symbol records which operation lies along which direction, and whether a given letter can appear without a centre attached is settled by whether its order is odd.

The same group, and not merely a similar one #

The replacement is an isomorphism, so a twisted class and its partner are the same abstract group. 4̅ and 4 are both cyclic of order four. 4mm, 4̅2m and 422 are all the dihedral group of order eight. 4̅3m and 432 are both the symmetric group on four letters, which is why a chemist uses one character table for the tetrahedral molecules and the octahedral ones, with the class of 4̅ operations standing where the quarter-turns stand and the mirrors where the diagonal half-turns stand.

Thirty-two classes, eighteen groups found isomorphisms like these by search — for each pair of classes, a hunt through images of a generating set for a bijection that preserves products — and reported that the thirty-two collapse to eighteen abstract groups. The theorem predicts that collapse without a hunt. Every twisted class is isomorphic to its partner by a formula, so the ten twisted classes can add no abstract group the proper ones lack. Every centred class is its rotations times a group of order two, so it adds a new abstract group only when no proper class already has that structure.

Counting kind by kind confirms the prediction. The eleven proper classes are eleven different abstract groups, the ten twisted classes add none, and the eleven centred classes add seven — mmm, 4/m, 4/mmm, 6/m, 6/mmm, m3̅ and m3̅m. The other four repeat a proper class: 1̅ is the same group as 2, 2/m as 222, 3̅ as 6, and 3̅m as 622. Eleven and seven are eighteen, and the search that found eighteen found nothing the construction does not account for.

What an isomorphism does not preserve is the action on space, and the action is what a crystal’s properties respond to. 4̅3m and 432 have identical multiplication tables and permit different physical properties. How different turns out to be computable from the rotation group alone.

Every property count, predicted from the rotations #

A physical property of a crystal is a tensor, and the number of independent components a class permits is the average, over the class’s operations, of the trace of each operation acting on the tensor’s components. A twist replaces some operations M′ by their negatives, so what it does to a count depends on one thing only: what the inversion does to that tensor.

The inversion commutes with everything, and on each kind of tensor it acts as a single sign. A dielectric tensor has two indices, each picks up a minus, and the tensor is unchanged. An elastic tensor has four and is unchanged. A polar vector has one index and reverses. The piezoelectric moduli have three and reverse. The gyration tensor behind optical activity has two indices and an extra factor of the determinant, because it describes a sense of rotation rather than a direction, and it reverses.

That sign is the parity of the property, and it decides everything. On an operation outside the kept half, which is the negative of an operation M′ of the partner, the property’s trace is the parity times its trace at M′. Two rules follow at once.

A property the inversion leaves alone has, in a twisted class, exactly the count its partner has. 4̅3m has three elastic constants because 432 has three; mm2 has three dielectric components because 222 has three.

A property the inversion reverses has the partner’s count with a sign attached: the average of the partner’s traces, weighted by +1 on the kept half and by −1 on the rest. That is usually a different number from the plain count, and computing it on the partner gives the twisted class’s count exactly.

All fifty counts agree with the prediction, and in twenty-four of the thirty cells for reversed properties the signed count differs from the plain one, so the sign is doing work rather than being carried along. Three rows repay reading.

432 against 4̅3m. 432 permits no piezoelectric modulus at all, and it is the one class without a centre that permits none. Its twist 4̅3m permits exactly one. The group is the same; the signed count over the same twenty-four rotations comes to one where the plain count comes to zero, because the twelve operations that are negated contribute with the opposite sign.

The polar classes. A spontaneous polarisation needs a direction every operation leaves alone, and ten classes have one. Five are proper — 1, 2, 3, 4 and 6 — and the other five are twists: m, mm2, 4mm, 3m and 6mm. Four of those five are twists of the dihedral classes 222, 422, 32 and 622, which have no polar direction whatever. Twisting turns their in-plane half-turns into mirrors that contain the principal axis, and a mirror containing a direction leaves it alone where the half-turn had reversed it, so the signed count is one where the plain count is zero.

Optical activity. Fifteen classes permit it: the eleven proper ones and four more, m, mm2, 4̅ and 4̅2m. All four extras are twisted classes, and in the table they are exactly the four rows whose signed gyration count is not zero. In the other six twisted classes the partner’s rotations permit a gyration tensor and the sign removes it.

The centred classes obey the same rule with a plainer outcome. Every operation outside their rotations is the negative of a rotation, so a property the inversion leaves alone has the rotations’ count and a property the inversion reverses has none. That is the familiar statement that a centre forbids piezoelectricity, pyroelectricity and optical activity, recovered as one case of three.

What the three kinds do not settle #

The classes here are geometric, not arithmetic. The theorem sorts groups of matrices up to a change of basis, so two twists that are one class on different axes are merged. A lattice separates them again. On a tetragonal lattice 4̅2m and 4̅m2 are different arithmetic classes, because the kept 222 lies along the cell edges in one and along the diagonals in the other, and a lattice can tell an edge from a diagonal. The fourteen halves are nearer the arithmetic count than the ten twisted classes are, and finishing that count needs the lattice, which nothing above uses.

The construction is not special to crystals. No step of the argument used a lattice, so the same three cases sort every finite group of motions of space that fixes a point — the infinite families and the icosahedral group included. The icosahedral rotations have no subgroup of index two, so they have exactly one improper relative, their product with the centre. Only the five crystallographic orders are computed here.

A count is a permission. Every number in the property table says how many components a class allows, and none says that any of them is non-zero in a real material. What the count does not say applies to the signed count exactly as it applies to the plain one.

What was computed, and how. For every class, its rotations and whether it contains the inversion; for every proper class, every subgroup of index two, by enumerating subgroups; each twist, checked to be closed under products and identified by its census of operation types; the isomorphism, checked product by product in all ten twisted classes; the abstract groups each kind contributes, checked against the separate count of eighteen; and every property count twice, once on the class and once from its partner. Two inputs are refused rather than reported. Negating the complement of a subgroup of index three does not give a group, and reading a centred class as a twist collapses its forty-eight operations onto twenty-four matrices.

Who put the pieces in this order #

The thirty-two classes were first enumerated by Johann Hessel in 1830, in a paper that went almost unread, and again by Axel Gadolin in 1867, whose stereographic drawings are the ancestors of the ones above. Both lists were lists of crystal forms before they were lists of groups.

The reduction to rotation groups belongs to the period after the classes had been recognised as groups. Once a mirror is seen as a rotation composed with the central inversion, the finite groups of motions of space divide at once into those containing the inversion and those that do not, and the second kind is a rotation group in disguise. Hermann Weyl sets the list out in exactly this way in Symmetry (1952), and it is the form in which the finite groups of the sphere are usually derived: the rotations first, then the centre, then the halves.

The chemist’s version of the isomorphism is older than the argument for it and was noticed from the tables rather than derived. Character tables for the tetrahedral and octahedral groups, and for C₃ᵥ and D₃, were printed identical except for the labels on their columns, and the labels were the only evidence that the operations were different.

Still open: the families past the five orders #

Everything above was computed at the five rotation orders a lattice permits, where the proper classes are finite in number. The theorem has no such limit. A cyclic group of rotations has a half whenever its order is even; a dihedral group has one half when its principal order is odd and three when it is even. So the twisted groups form infinite families alongside the rotation groups, and together with the centred ones they make seven families of groups with a single principal axis. Which seven, and why they number the same as the ways a pattern can repeat along a strip, is a question with a geometric answer rather than an algebraic one — and the answer is to wrap the strip round a cylinder.