Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Symmetry is decidable

Unusually for a physical science, the central questions here have exact answers computable in integer arithmetic. There is no tolerance to choose and no residual to interpret.
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 hexagonal lattice. 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

The lattice underneath

Strip a pattern of everything but its repeats and a grid of points is left. That grid is not decoration — it is the object that decides which symmetries the pattern is permitted to have.

 The angles between the faces of {102̅}. The form {102̅} of class 3̅m in section, with each face labelled by its indices. Its 6 faces make 15 pairs and only 3 distinct angles, the smallest being 76.43°. Every value is computed from the cell's metric — the one calculation in this family that is not integer arithmetic, because an angle is a real number and a lattice does not constrain it. Symmetry at work

Why a crystal face carries small whole numbers

A crystal face is flat because it lies on a plane of lattice points, and its orientation is therefore named by three integers rather than by two angles. That the integers exist is a theorem about lattices; that they are usually smaller than four is a separate claim about growth, and the two are routinely run together.

 A lattice and its reciprocal. The reciprocal lattice is where a crystal scatters. A long axis in the crystal gives closely spaced spots and a short one gives widely spaced spots, so the diffraction pattern is the structure turned inside out. How it is known

The reciprocal lattice

Nobody has seen a space group. Crystals are read from where they scatter, and where they scatter is a second lattice in which long has become short and short has become long.

 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.

 A rotation. The motif in the first colour, its images under a single rotation in the second, and the symmetry element marked where the operation itself says it lies. Operations

What a symmetry actually is

Not a property of a shape but a motion that leaves it alone. Once symmetry is a verb rather than an adjective, everything else in the subject follows — including why there can only ever be seventeen wallpapers.

 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.

 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.

 Two habits of {101̅}, one set of angles. The same 12 faces of class 6/mmm, grown to different distances from the centre. The outline changes completely and not one interfacial angle moves, because a face's orientation is set by the lattice and its extent by how fast it grew. That is Steno's law, and it is the reason a goniometer measures something about the substance rather than about the specimen. Symmetry at work

The angles belong to the substance, the shape to the specimen

Two crystals of the same mineral can look nothing like each other and still have exactly the same angles between corresponding faces. That is the oldest quantitative law in the subject, and what it measures turns out to be the shape of the unit cell — which means a brass instrument from 1809 was reading lattice parameters a century before anyone knew there were any.

 A glide. The motif in the first colour, its images under a single glide in the second, and the symmetry element marked where the operation itself says it lies. Operations

The four motions of the plane

Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.

 The round trip, on Pnma. 8 operations were generated from the standard generators of Pnma; the orbit of three points in general position was formed, the group was discarded, and 8 operations were rediscovered from the 24 points alone. The two sets are identical, which is what the figure asserts. Into space

Forgetting a group in three dimensions

For three phases this site said its machinery was two-dimensional and decided nothing about a space group. That was true, and it was a limit rather than a principle — nothing in the decidability argument mentions the number two.

 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 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. Into space

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.

 Two cells of equal area on one rhombic lattice. One lattice — the rhombic lattice that cm sits on — with two parallelograms drawn on it: the conventional cell, and a sheared cell whose edges are the integer combinations (1, 0) and (1, 1) of it. The points are identical in both outlines; only the description changes. Each cell's contents were counted by writing every lattice point in that cell's own coordinates and sharing each one out between the cells that meet at it — a quarter at a corner, a half on an edge, one inside — and the totals come to 1 and 1, which are the determinants of the two matrices. The alternative has determinant one, so its inverse is integral and it generates exactly the same lattice; that is the whole condition, and it is why a lattice has infinitely many bases and no arithmetic can prefer one. Lattices

The cell is a choice, the lattice is not

Every lattice has infinitely many unit cells and infinitely many bases, and crystallography picks one by convention. Knowing which convention is in force is the difference between a symbol that means something and a symbol that means nothing.

 The phase problem. The same structure rebuilt from its diffraction three ways: with the true amplitudes and phases, with the phases scrambled, and with the amplitudes discarded but the phases kept. The atoms survive the loss of the amplitudes and do not survive the loss of the phases, which is what an experiment throws away. How it is known

The phase problem

A detector records how much light arrives and not when it arrives, so half of every diffraction measurement is thrown away before it is written down. The half that is lost turns out to be the half that carries the structure.

 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.

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