Before the lattice has a say

The crystallographic restriction is usually met as a prohibition: a lattice admits rotations of order 1, 2, 3, 4 and 6, and nothing else. That framing hides a prior question. Before any lattice is involved, what are the finite symmetry groups at all? — and the answer is a short, complete list that has nothing crystallographic in it.

In the plane the answer is two families and the proof fits in a paragraph. In space it is five families, and the proof is a Diophantine equation with four solutions. The restriction then arrives and deletes most of the list, and what survives is eleven groups. Reaching the crystal classes that way — by deleting from a complete list rather than by searching inside a holohedry — makes it obvious which of them are there because of the lattice and which would be there anyway.

Leonardo’s theorem, and why the centroid is the whole proof #

A finite group of isometries of the plane fixes a point. That is the entire content, and the argument is a trick that generalises to almost everything.

Take any point and form its orbit — the finite set of images under the group. Every element of the group permutes that set, because applying it to an image gives another image. A permutation of a finite set of points leaves their centre of mass where it was. So every element of the group fixes the centroid of the orbit.

An isometry of the plane fixing a point is a rotation about it or a reflection through it. A finite group of rotations about one point is cyclic, since the smallest positive rotation in it generates the rest. Adding any reflection doubles the group and gives the dihedral case. So the list is Cₙ and Dₙ, and there is nothing else — a result attributed to Leonardo da Vinci, who wanted to know which arrangements of chapels could be attached to a central plan without spoiling its symmetry.

Counting poles: the equation that classifies the sphere #

Space is harder and the classification is more surprising. The standard argument counts one thing two ways, which is the standard way to get an equation out of nothing.

Let G be a finite group of rotations of order N. Every non-identity rotation fixes exactly two points of the sphere — the two ends of its axis — and those fixed points are called poles. Count the pairs (rotation, pole it fixes): there are 2(N − 1) of them, since each of the N − 1 non-identity rotations contributes two.

Now count the same pairs by pole. The poles fall into orbits under G; a pole in an orbit of size m has a stabiliser of order N/m, which contributes N/m − 1 non-identity rotations. So the total is Σ over orbits of m(N/m − 1) = Σ (N − m).

Setting the two counts equal and dividing by N gives

$$2 - \frac{2}{N} = \sum_i \left(1 - \frac{1}{n_i}\right)$$

with nᵢ the order of the i-th class of axes. Every term on the right is at least ½; the left side is less than 2; so there are at most three classes of axes, and the classification has become a search over small integers.

The shape of the answer is worth pausing on. Two infinite families, and then three isolated groups that exist for no reason expressible in the equation — they are what is left when the arithmetic happens to work out. The largest is the icosahedral group of order sixty, which will turn out to be the one a crystal may never have.

Built, and counted, and required to agree #

A search over an equation gives symbols. The site’s habit is to build the objects the symbols name and count them independently, and the two answers have to match.

Each family is constructed as explicit matrices: the cyclic and dihedral ones from rotations about an axis, the octahedral one as the proper subgroup of the cubic holohedry, the tetrahedral one by closing two of its elements, and the icosahedral one from the twelve vertices of an icosahedron. Then every rotation’s axis is found from its matrix, the axes are collected up to sign, and the census is compared with the solution.

The octahedral case is the one where the construction and the equation are most obviously doing different things. The equation says (2, 3, 4) at order 24 and stops; the construction produces twenty-four explicit integer matrices, and the census sorts them into six twofold axes, four threefold and three fourfold. Those numbers are not in the equation — it names the orders of the classes and not how many axes each class holds — so the census is genuinely new information, and it is what a reader needs in order to recognise the group in a crystal.

What the equation is really counting #

The pole-counting argument is worth one more paragraph, because it is the same argument as several others on this site wearing a different hat.

What it counts is orbits with their stabilisers, which is the Burnside relation in the form a geometer meets it: the sum over orbits of (1 − 1/|stabiliser|) is fixed by the order of the group. The same accounting turns up in the fundamental domain, where the area of the domain is the cell’s area over the order, and the special positions are the points whose stabiliser is bigger than the identity; and it turns up in orbifold notation, where the “cost” of a cone point of order n is 1 − 1/n and the seventeen plane groups are the ways of making the costs add to exactly two.

The correspondence is exact rather than an analogy. Orbifold notation’s magic-theorem cost of 2 is the flat case; the sphere’s finite groups are the cases where the cost is less than two, and the deficit is 2/N; and the hyperbolic case, with infinitely many groups, is the cost exceeding two. One equation, three geometries, and the finite groups are the ones whose orbifold is a sphere.

The cut, and the eleven #

Now the lattice arrives. A group is the point group of a crystal exactly when a lattice admits it, and by the crystallographic restriction that means every one of its axis orders lies in {1, 2, 3, 4, 6}.

Applying that to the list is a deletion, and the deletion is brutal. Every Cₙ and Dₙ with n outside the five goes: C₅, C₇, C₈, D₅, D₇ and infinitely many more. The tetrahedral and octahedral groups survive, because their axes are of orders 2, 3 and 4. And the icosahedral group is deleted, by its fivefold axes and by nothing else.

The deletion has a shape worth stating in one line, because it is the shape of the whole subject. Of the two infinite families, a lattice keeps five members of each and deletes the infinite tail; of the three sporadic groups it keeps two and deletes the largest. So periodicity does not trim the list — it truncates two families and removes the most symmetric object on it, which is why crystals look the way they do and why the exceptions are so conspicuous when they occur.

Eleven is a number this site has met twice already, in different clothes. It is the count of enantiomorphic crystal classes — the classes with no improper operation, which is exactly what “group of rotations” means. And it is the number of Laue classes, which is a different eleven arrived at by a different argument and is worth not confusing with this one.

The duplicate is instructive rather than untidy. An enumeration that produces the same object twice under two names is the commonest failure this site has met — the twenty-eight classes that came from reading an order off a matrix, the ten arithmetic classes that started as sixty-four — and finding the duplicate is part of the enumeration rather than tidying up afterwards. Here the test is the axis census: same order, same axes, same group.

Why the deletion falls where it does #

There is a reason the two polyhedral survivors survive and the third does not, and it is not that five is unlucky.

The tetrahedral and octahedral groups are subgroups of the symmetry group of a cube, and a cube is built on a lattice. The icosahedral group is not a subgroup of any lattice’s symmetry group, because it contains a fivefold rotation and no lattice does. So the survivors are exactly the ones that were compatible with periodicity to begin with, which sounds circular and is not: the equation knew nothing about lattices, produced three sporadic groups, and the restriction happened to keep two of them.

The one it deletes is the one nature keeps producing anyway. Virus capsids are icosahedral; boron cages are icosahedral; quasicrystals have icosahedral diffraction patterns. What Shechtman measured was a group from this list that the restriction had ruled out of crystallography, appearing in something that diffracted sharply — and the resolution was that the object was not periodic, so the restriction never applied to it.

Who established it, and in what order #

The plane case is old and its attribution is a courtesy. Leonardo’s notebooks contain the enumeration of central plans with chapels attached, which is the classification of finite plane groups in architectural dress; the mathematical statement waited until group theory existed to state it in.

The spherical case belongs to the nineteenth century and to Felix Klein, whose Lectures on the Icosahedron of 1884 made the classification of finite rotation groups a piece of standard equipment — and did so in service of a question about quintic equations rather than about symmetry. The pole-counting proof in the form above is essentially Klein’s, and it is still the shortest one known.

The crystallographic cut is later still and belongs to the mineralogists: Hessel enumerated the thirty-two classes in 1830 from the shapes of crystals, Gadolin rederived them in 1867, and neither of them had Klein’s list to delete from. So the historical order is the reverse of the logical one — the crystal classes were known for fifty years before the classification they are a subset of, and the fact that the subset is exactly the one the restriction picks out was noticed afterwards.

Five solids, three groups #

The polyhedral entries in the list are usually introduced by naming the Platonic solids, and the count does not match: there are five solids and three groups. The mismatch is the useful part.

A polyhedron and its dual — the solid whose vertices sit at the first one’s face centres — have the same symmetry, because the construction that produces one from the other is built entirely out of the first one’s own symmetry. The cube and the octahedron are dual; the dodecahedron and the icosahedron are dual; the tetrahedron is self-dual.

So the five solids fall into three symmetry types, and those three types are exactly the three polyhedral entries the pole-counting equation returns. The equation knew the duality before anybody mentioned polyhedra: it produces the triples (2,3,3), (2,3,4) and (2,3,5), one for each type, and it produces no fourth. That the solids come in five and their groups in three is not an accident of Greek geometry but the same statement the arithmetic makes.

It is also where the equation’s second column comes from. Read the triples as what the axes are: (2,3,4) has axes of order four through the cube’s faces, three through its vertices, and two through its edge midpoints — and through the octahedron’s vertices, faces and edges respectively, which is the duality visible in the reading rather than in the drawing.

The same equation, one line up and one line down #

The equation classifying the sphere is not confined to the sphere, and recognising it in its other two settings is the shortest route to seeing why the list is finite.

Written with the left side as 2 − 2/N, the equation says a certain sum of terms 1 − 1/nᵢ comes to slightly less than two. Each term is a price, the total is a budget, and the budget is under two by an amount that shrinks as the group grows. That is the orbifold accounting run below two — a total under two buys a finite group acting on a sphere, and the arithmetic here is where the finiteness comes from.

At exactly two the budget buys a group acting on the plane, and the answer is the seventeen. Above two it buys a group acting on the hyperbolic plane, and there are infinitely many. One equation, three geometries, and the sign of a single quantity deciding which.

That framing explains a feature of the list above that otherwise looks arbitrary: why the polyhedral entries stop at (2,3,5). The next candidate, (2,3,6), makes the sum exactly two — no N satisfies the equation, because the left side is strictly less than two for every finite N. So (2,3,6) is not a finite group at all; it is p6m, one of the seventeen, and the reason it is missing from a list of finite groups is that it is infinite. The three geometries meet at that triple.

It is worth stating what that meeting-point buys, because it is more than an amusing coincidence of arithmetic. The three geometries are usually presented as three subjects — finite groups, wallpaper groups, Fuchsian groups — with three literatures and three vocabularies. One equation covers all three, and the only thing that changes between them is the sign of a rational number. So a reader who has followed the pole count here has already done most of the work for the flat case and for the hyperbolic one, and the eleven that survive the lattice’s cut are a sublist of a sublist rather than a separate result.

The deletion also runs one way only. Every group on the crystallographic list is a finite rotation group, so nothing survives the cut that was not on the list to begin with; but a group deleted by the cut loses none of its standing as a group. The icosahedral group is exactly as real after the restriction as before it, and the century of confusion about what a five-fold diffraction pattern could mean was a confusion about which list a measurement was evidence against.

Where the exactness stops #

Three limits worth naming.

These are rotation groups. Adding improper operations — reflections, rotoinversions — makes the full list of finite subgroups of O(3), which is longer: fourteen families rather than five, since each rotation group gives rise to several by different combinations with the inversion. The eleven above are the proper crystal classes; the thirty-two come from doing the same work with improper operations admitted, which this site does from the other end.

The equation classifies up to conjugacy in the rotation group, not up to isomorphism. D₃ and C₆ both have order six and are different groups; C₂ and D₁ have the same order and are the same group. Neither statement is decided by the order, and the census is what decides them.

A group’s presence on the list is not a claim that anything has it. C₇ is a perfectly good finite group and no crystal has it; so is C₁₇. The list says what is available to a bounded object, and a great deal of what is available never occurs — which is the difference between a classification and a census, and the reason counting the groups ornament actually uses is a different exercise from counting the groups.

And “finite” is doing all the work. The centroid argument needs a finite orbit, and the pole-counting argument needs a finite group. Infinite groups of isometries — the plane groups, the space groups, the rod and layer groups — are classified by completely different arguments, and none of the lists above says anything about them.

Where the ladder goes next #

The list above is what symmetry offers before periodicity is imposed. The restriction then deletes the fivefold groups from crystals — and a crystal can still contain one locally, in a cluster or a molecule or a virus, provided the lattice never has to reproduce it. That is non-crystallographic symmetry, it is exact where it applies, and it is the next rung.