Crystallography & Symmetry
patterns derived from their groups
Concept

Symmorphic — where it appears

Said of a space group having an origin at which every operation's translation part vanishes, so that it holds no screw and no glide. There is exactly one such group per arithmetic crystal class, which is why there are seventy-three of them and seventy-three classes.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

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.

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.

space-groups · Space groups 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 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.

classification · Seventeen 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.

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.

classification · Seventeen The translation of every operation, split in two. Every operation of P2, P2₁, Pm and Pc other than the identity, with its translation split into the intrinsic part — one n-th of the sum of the operation applied to itself n times, which no choice of origin can remove — and the location part, which is only a statement about where the origin was put. 2 of the 4 operations shown have a non-zero intrinsic part, and those are exactly the screws and the glides.

The half of a translation that is not a choice

Every operation's translation splits in two — a part that belongs to the operation and a part that only records where somebody put the origin. Almost everything peculiar about space groups is a consequence of that split, including why there are two hundred and thirty rather than seventy-three.

space-groups · Space groups The seventeen sorted by lattice: 2, 5, 2, 3, 5. The five plane lattices, each drawn from the basis every other figure here uses, with the wallpaper groups that sit on it and the order of each against its lattice's holohedry. The counts are 2, 5, 2, 3, 5, which is seventeen again, arrived at by a different route from the case analysis on rotation order. Two relations hold and both are checked. Every group's order divides its lattice's holohedry, because an operation has to map the lattice onto itself before it can map the pattern onto itself — which is why a quarter turn has nowhere to live but a square lattice. And the converse fails on every one of the five: each lattice carries at least one group whose order falls short of what the lattice offers, so knowing the lattice narrows the group to a handful of candidates and never to one. The pairs printed in the accent colour are the groups that take everything their lattice permits.

The classification proof, one branch at a time

Seventeen is a theorem, and the argument that establishes it is a finite case analysis that fits on a few pages. Working through it is the difference between knowing the number and knowing why there is no eighteenth.

classification · Seventeen Each arithmetic class holds exactly one symmorphic group. The arithmetic crystal classes counted here in full, with the number of space groups each produces and the number of those that are symmorphic — that is, that have an origin at which every operation's translation part vanishes. The right-hand column is one in every row, over 25 groups in all, and each one is computed from the class, not copied from a table. That is the bijection behind the number 73: there are seventy-three arithmetic crystal classes in three dimensions and seventy-three symmorphic space groups, and the correspondence is this one, class by class.

One symmorphic group per class

Every arithmetic crystal class holds exactly one space group in which some origin clears every translation part at once. That bijection is why there are seventy-three symmorphic space groups and seventy-three arithmetic classes, and it is checked here class by class rather than counted.

space-groups · Space groups Thirteen ways to hold a lattice. Every finite group of integer matrices in two dimensions, up to a change of integer basis: 13 of them. Ten different abstract groups appear, and three of the ten hold a lattice in two inequivalent ways — a mirror along an axis or along a diagonal, and the same for 2mm and for 3m. The enumeration is a search: every subgroup of the two maximal holohedries, merged by conjugacy under integer matrices of determinant ±1, with the answer checked for not depending on how wide the search was.

Thirteen ways to hold a lattice

The crystallographic restriction is about one matrix. A crystal has a whole group of them acting on one lattice at once, and asking how many such groups there are gives thirteen — not the ten of the plane point groups, and not the seventeen of the plane groups.

restriction · Restriction p4g in 4 letters and 8 relations. The presentation of p4g, derived from the group's own operations. The two translations commute; each conjugation relation is read off a column of a matrix; and the point group's relations are corrected by the translation they actually come back as, which is what makes this group an extension rather than a semidirect product. Every relator is evaluated where the group lives and must be the identity, and coset enumeration on the letters alone returns 8, which is the order of the point group.

A group in four letters

Every other essay here describes a symmetry group by what it does to the plane. There is a second description — a handful of letters and the words in them that are required to equal nothing — and it can be counted with no plane anywhere in the computation.

operations · Presentations pg scatters as pmm. The orbit of a motif under pg, and the set of all ordered differences between its points brought to a common origin. The vector set was handed to the detector with no indication of where it came from, and came back as pmm: 31 peaks from 6 atoms, on the same lattice.

Seventeen groups, seven vector sets

A map of interatomic vectors is more symmetric than the structure it came from, twice over: it always acquires a centre, and it loses every translation part. So a glide becomes a mirror, seventeen plane groups collapse onto seven — and the collapse is verified by handing the vectors to a detector that has never heard of Patterson.

diffraction · The Patterson function How the space groups are spread over their classes. For each number of groups, how many arithmetic classes carry exactly that many space-group types, in space and in the plane, every count derived from the classes' cohomology and the normaliser's action on it. In the plane ten of the thirteen classes carry one group. In space twelve of the seventy-three carry one, thirty-five carry two and fifteen carry four, so the typical class carries two; the long tail is the orthorhombic and tetragonal classes with eight, ten and sixteen.

Two hundred and nineteen over seventy-three

The space groups average three to an arithmetic class, and an average of three could be a few classes carrying dozens or every class carrying a handful. Derived class by class from the cohomology and its normaliser, the answer is the second: twelve classes carry one group, the median carries two, and the largest carries sixteen. Sixty-one of the seventy-three admit a non-symmorphic group, and the step that merges relabellings, which the plane barely uses, removes more than a quarter of the list.

restriction · Finiteness I222 and I2₁2₁2₁: the same axes, apart. The two-fold rotation axes of the two space groups on the body-centred class of 222, drawn in one conventional cell seen down its third edge: axes along that edge as diamonds, axes along the first edge as solid lines and along the second as dashed lines, with the height of each written beside it. Both groups have pure two-fold rotation axes in all three directions. In the symmorphic group some axes of different directions cross, at the origin among other places; in the other none does, because the heights of the axes in each direction are a quarter out of step with the positions of the axes in the others.

Two groups no single operation reveals

A space group is non-symmorphic when no choice of origin removes all of its translations at once, and every textbook example has a screw or a glide to show for it. Two of the two hundred and thirty do not need one. Every operation of I2₁2₁2₁, taken on its own, is also a plain rotation about some axis; so is every operation of I2₁3. What makes them non-symmorphic is that their rotation axes never meet — and no diffraction extinction can tell them from I222 and I23.

classification · Cohomology Two hundred and twelve of the 219 are made of their own operations. The 219 affine space-group types divided by which of their own operations generate them, lattice translations included. 110 are generated both by their screws and glides alone and by their rotations, mirrors and inversions alone; 67 by their screws and glides only; 29 by their point operations only; 6 only by the two kinds together; and 7 by nothing they contain — P1, P2, Pm, Pmm2, P4, P3, P6 — which need a translation given outright.

Seven space groups cannot make their own translations

Every space group contains a lattice of translations, and for all but seven of the 219 types those translations are products of the group's other operations. A screw done twice climbs its axis and a glide done twice slides along its plane, so the non-symmorphic operations make lattices by themselves. The seven that cannot are exactly the seven plane groups with no glide, repeated along a line they all fix, and the reason is a difference between a glide line and a glide plane.

space-groups · Screws and glides Two hundred and thirty groups on a hundred and twenty-two diffraction symbols. Every space group placed in a column for its diffraction symbol — Bravais lattice, Laue group and systematic absences — with the columns grouped by lattice. There are 122 symbols. 50 of them name one group; 44 are shared by two, 23 by three, 2 by four and 3 by five.

Every ambiguity in a diffraction pattern, sorted by its cause

A diffraction pattern shows a lattice, the symmetry of its intensities and the reflections that are missing, and the 230 space groups show 122 different combinations of the three. Fifty of those name a single group. For the rest, each pair of groups that share one can be sorted by why they do, and there are five reasons. Four of them are ambiguities some further measurement resolves. The fifth, two groups with the same operations in the same places except for where their axes cross, happens exactly twice, and nothing but a solved structure decides it.

classification · Cohomology

Named alongside it

The objects these essays reach for when they reach for this one.

Point groupArithmetic crystal classScrew axisGroup extensionClassificationGlide planeIntrinsic translationSpace groupSystematic absenceCocycleEnantiomorphEnumeration

All concepts