Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Exactly this many

Seven friezes, seventeen wallpapers, fourteen lattices, two hundred and thirty space groups. Each number is a theorem with a finite proof, and the word that matters in every one of them is 'exactly'.
P2₁/c, in the two diagrams the Tables print. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness. Into space

The step a flat surface has no room for

Seventeen becomes two hundred and thirty, and the factor of thirteen is not a matter of having more directions to work in. It comes from an operation the plane cannot hold — a translation that no choice of origin will remove.

 The five rotations a lattice will carry. One motif and every rotation a plane lattice permits: orders 1, 2, 3, 4, 6, and nothing else up to 12. Each panel turns the motif by its own operation as many times as the order allows, on the lattice that operation requires — oblique for the identity and the half turn, hexagonal for the third and the sixth of a turn, square for the quarter. The trace printed under each is the sum of the diagonal of the operation's matrix written in the lattice's own basis, and it is a whole number in every panel, which is the entire content of the crystallographic restriction. The list of orders is produced twice, once from that trace condition and once from the degree of a cyclotomic polynomial, and the figure refuses to draw if the two disagree. What a lattice forbids

The crystallographic restriction

A repeating pattern may have rotations of order two, three, four or six, and nothing else whatever. The proof is one line of arithmetic, and everything finite in the subject descends from it.

 The seventeen wallpaper groups. Every way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others. The classification

The seventeen

Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon.

 The thirty-two crystal classes. Every crystallographic point group, as a stereogram. Each was found by enumerating the subgroups of m3̅m and of 6/mmm, and each diagram is the orbit of one general direction under the group, filled where the pole is in the upper hemisphere and open where it is in the lower — which is the only thing in the picture that tells a rotation from a rotoinversion. What symmetry decides

Thirty-two, and no others

There are exactly thirty-two ways a crystal can be symmetric about a point. Not thirty-two that anybody has catalogued — thirty-two that a finite search produces, from two starting groups, with every step of the reduction counted separately so that no two of them can quietly compensate.

 The five plane lattices. Every periodic pattern in the plane repeats on one of five lattices. The classification is by which point symmetries the lattice itself admits, and the five are exhaustive — a sixth would need a rotation order no lattice can carry. Lattices

Five lattices, and no others

A repeating grid can be oblique, rectangular, centred, square or hexagonal. That is the complete list for the plane, and the argument that closes it is a page long.

 What pg scatters. The diffraction pattern computed from the atom positions alone. Where a glide plane is present, alternate reflections along a row cancel exactly — and those missing spots are how the glide is identified in an experiment, since the glide itself is never seen. How it is known

Systematic absences

The most informative part of a diffraction pattern is the part that is not there. A glide plane cancels alternate reflections along a row, exactly, and those missing spots are how a symmetry nobody can see is identified.

 The merge, and what witnesses it. The two holohedries are enumerated separately and their class lists merged where the element signatures agree. That returns the right total, which is not the same as being right: 33 merges are made and every one of them is checked by constructing an explicit change of basis carrying one group onto the other. What symmetry decides

A fingerprint that gave the right answer

The thirty-two classes were merged on a fingerprint — the census of operation types — and the fingerprint returned thirty-two, which is correct. Returning the correct answer is not the same as being entitled to it, and the difference took three wrong constructions to close.

 One tile becomes 130 in 3 inflations. One tile subdivided into smaller copies of the same two shapes, repeatedly. The rule is local and deterministic, and the pattern it builds has long-range order without any repeating cell. The panel counts are 10, 20, 50, 130, and each is the one before it multiplied by the substitution matrix — an identity in whole numbers, checked every time the figure is drawn rather than quoted. Order without repetition

Inflation, and where the golden ratio comes from

A Penrose tiling is not laid out tile by tile. It is grown by cutting every tile into smaller ones and rescaling, and the growth matrix's dominant eigenvalue is the golden ratio squared.

 The wallpaper group p4m. A pattern with the symmetry of p4m, generated by applying the group's 8 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly. The classification

Reading Hermann–Mauguin

p4g looks like a licence plate and is in fact a set of instructions. Half an hour with the rules turns the seventeen from a list to be memorised into a notation that can be read.

 Assuming a 5-fold rotation. The shortest lattice vector, its rotated copies, and the combination of them that is itself a lattice vector. Where that combination comes out shorter than the vector assumed shortest, the assumed rotation cannot exist. What a lattice forbids

Why five-fold is impossible

A second proof, geometric rather than algebraic: assume a five-fold centre, and out of it construct a lattice vector shorter than the shortest one there is.

 The wallpaper group p3m1. A pattern with the symmetry of p3m1, generated by applying the group's 6 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly. The classification

p3m1 and p31m

Two groups with the same lattice, the same point group and the same number of operations, differing only in where the mirrors sit. The pair is the clearest evidence that position is as much a part of a symmetry as presence.

 {111} in class m3̅m. The form {111} of crystal class m3̅m: 8 faces, being the orbit of one face under the 48 operations of the class, with a stabiliser of order 6. 4 poles lie in the upper hemisphere or in the plane of the page and are drawn filled; the other 4 lie below and are drawn open at the same positions, which is the stereographic convention and the reason only the upper ones carry their indices. The form is closed: the faces enclose a volume, so a crystal can be bounded by this form alone. Symmetry at work

A form is an orbit, and whether it closes is an integer question

Name one face of a crystal and its class names the rest. That set is a form, it is an orbit in exactly the sense this site has used since its first essay, and whether it encloses a volume — whether a crystal could be bounded by it alone — is decided without any lengths or angles entering the calculation anywhere.

 The 48 point symmetries of a cubic lattice. Every operation that maps a cubic lattice onto itself, built as the integer matrices preserving that system's metric: 48 of them, of which 24 are proper rotations and 24 reverse handedness. The orders occurring among the rotations are 1, 2, 3, 4 — the same list the plane gives, so the crystallographic restriction does not change in three dimensions, and there is no six anywhere. The 13 rotation axes are counted from the rotations they carry rather than drawn from memory, and every rotation but the identity is checked to belong to exactly one of them. What a lattice forbids

The restriction in three dimensions

Space is roomier than the plane in every other respect, so the natural expectation is that it permits more rotation orders. It permits exactly the same five, and seeing why is more interesting than the result.

 The crystal classes 4, 4̅, 3̅, 6̅. 4, 4̅, 3̅, 6̅: the orbit of a general direction under each group, giving 4, 4, 6, 6 poles, with general positions and symmetry elements. Filled marks are poles above the plane of the page and open ones below it. What symmetry decides

What a trace decides

Ten kinds of operation, and two integers tell them apart. The determinant and the trace name a symmetry operation completely — which is why the classification can be run on integer matrices in a lattice basis and never once ask what angle anything turns through.

 Doing one after another. Two symmetries of a pattern, and the one that doing both lands on. The third picture is not a new operation drawn to fit — it is the composition, and it was already in the group. Operations

Why it is a group and not a list

The symmetries of a pattern cannot be chosen independently. Do two of them in succession and the result is forced to be a third, which is why there is no eighteenth wallpaper for anybody to invent.

 Reading 4/mmm off its own directions. Each position of 4/m2/m2/m reports one symmetry direction of the tetragonal system: the highest-order axis lying along it, and whether a mirror is perpendicular to it. Nothing is looked up — every row is computed from the group's own matrices. What symmetry decides

Reading a class off its own axes

A Hermann–Mauguin symbol is not a name that was assigned. It is a report on three directions, read in order, and the whole of it can be derived from the group's matrices — with one genuine convention and one exception, and the exception is orthorhombic.

 The space groups in class mm2 (P), counted. Every way of attaching translations to the generators of mm2 (P): 64 assignments close into a group of the right size, 16 survive moving the origin, and 10 survive relabelling the axes — which is the number the International Tables record for this class. the ten primitive orthorhombic groups with a polar axis. Into space

Sixteen candidates, ten groups

One point group, one lattice, and every consistent way of attaching translations to it — enumerated in full. The count comes out at sixteen, and then at ten, and the step between the two numbers is a decision about what "the same group" means rather than an arithmetic fact.

 168 reflections, 26 rings, one axis. Every reflection of a square lattice out to indices of 6, drawn as a point of the reciprocal lattice with the rings of equal length it lies on. A single crystal gives each point its own place on a detector; a powder is every orientation at once, so what is measured is the ring rather than the point, and the 168 reflections arrive as 26 lines on the single axis at the right, with height the number sharing each. The rings are found by sorting the lengths and grouping equal ones, and every point is checked to sit on the ring it was given. 1 of the rings are marked: they hold reflections that no operation of the lattice's point group relates, so the collapse has added together two genuinely independent measurements and no experiment can take them apart again. How it is known

What a powder pattern loses

Grind a crystal up and every orientation is present at once, so a two-dimensional pattern of spots collapses onto a single axis. Reflections that had their own places arrive together, and some of the coincidences are exact and have nothing to do with symmetry.

 The seven frieze groups. Every way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another. The classification

Seven friezes

The same classification argument on a strip instead of a plane, where it is short enough to check by hand. Seven ways to repeat a motif along a line, with names like hop, step and sidle.

 Every solution of the axis equation. The integer solutions of 2 − 2/N = Σ(1 − 1/nᵢ), which is what counting the pairs (rotation, fixed pole) two ways gives. Two classes of axis force n₁ = n₂ = N and give the cyclic groups; three classes give the dihedral family and exactly three sporadic answers — (2, 3, 3), (2, 3, 4) and (2, 3, 5), of orders 12, 24 and 60, which are the rotation groups of the tetrahedron, the octahedron and the icosahedron. Four classes are impossible, because four terms of at least a half already exceed the left-hand side. Nothing about crystals has been used. What a lattice forbids

Before the lattice has a say

Every finite group of motions of the plane is a Cₙ or a Dₙ, and every finite group of rotations of space is one of five families. Both lists come out of counting rather than out of crystallography — and then the crystallographic restriction deletes almost all of them, leaving eleven.

 The crystal classes 3m, 3̅m, 6̅2m. 3m, 3̅m, 6̅2m: the orbit of a general direction under each group, giving 6, 12, 12 poles, with general positions and symmetry elements. Filled marks are poles above the plane of the page and open ones below it. What symmetry decides

3m1 and 31m are one class

This site has an essay arguing that p3m1 and p31m are genuinely different groups. As point groups the same two objects are one class — and the two subgroups are each normal in the hexagonal holohedry, so nothing in the lattice relates them. What does is a rotation of thirty degrees.

 Reducing a basis. An awkward basis and the reduced one Gauss's algorithm returns. Both describe the same lattice — the change of basis has determinant one — and the reduced pair is the shortest vector together with the shortest independent of it, checked against an exhaustive search. Lattices

Reduction, and the shortest basis

Every lattice has infinitely many bases and no arithmetic picks a preferred one — until a rule is imposed. Reduction is that rule, it terminates in a handful of steps, and it is what lets a database decide whether two reported crystals are the same crystal.

 2 screw axes. 2 of the eleven screw axes a lattice permits, each drawn as the helix it is: 3₁, 3₂. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs. Into space

Eleven groups that are their own reflection's rival

There are two hundred and thirty space groups, and there are two hundred and nineteen. Both numbers are correct and they answer different questions, and the eleven that separate them are the reason a crystal can be built one way round and not the other.

 A conjugacy class of p4m. One conjugacy class of p4m drawn in place: every copy of the same symmetry that the group can carry onto every other. Conjugation was applied to each of the 8 operations by each of them in turn, and the kind and order of the result was checked to match every time. Operations

The same symmetry, somewhere else

Two mirrors in a pattern can be the same symmetry or two different ones, and looking will not settle it. Conjugation is the operation that decides, and it turns an intuition about sameness into arithmetic.

All themes · All essays