Theme

The theme: Exactly this many — page 2

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'.

Lattices

Centring, counted as a sublattice

Adding the centre of every cell to a lattice produces another lattice, containing the first with index two. Doing it to each of the five in turn shows why the list is five rather than ten, and why only one of the five has a centred description worth keeping.

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

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.

Building the reciprocal lattice from spacings. Each family of lattice rows has a spacing, and each contributes one reciprocal point: perpendicular to the rows, at the inverse of the spacing. The points built that way were compared against the algebraic definition and agree exactly.

Lattices

The dual lattice, as a construction

The reciprocal lattice is usually introduced as a formula and then used as a fact. Building it instead — one point per family of lattice rows, at the inverse of the spacing — makes every property it has obvious rather than memorable.

The substitution, 6 generations. The rule "every long tile becomes a long and a short, every short tile becomes a long", applied 6 times from a single tile. Each generation is as long as the previous two together, so the tile counts are Fibonacci numbers — 13 long and 8 short at the last row — and their ratio is 1.62500 against the golden ratio's 1.61803. The sequence never repeats and every finite piece of it recurs infinitely often, which is order without periodicity in its smallest form.

Order without repetition

The smallest quasicrystal

Two tile lengths on a line, in the golden ratio, in a sequence that never repeats. Three completely different constructions produce it, they are required here to agree, and its diffraction needs two integers per peak where a periodic chain needs one.

The thirty-two, by crystal system. 32 classes in 7 crystal systems. Each column is one crystal system and each cell one class, ordered by the number of operations it holds. Nothing here is tabulated: the classes come from the enumeration and the marking from a character sum over each group.

What symmetry decides

The holohedry is the ceiling

A crystal never has more point symmetry than its lattice. That single containment decides which system a class belongs to, why there are seven systems and not thirty-two, and why a lattice can be more symmetric than the crystal sitting on it — which is the usual case rather than the exception.

Which classes can twin by merohedry, and how many ways. Every crystal class, with the number of twin laws its own lattice offers it. The index of the class in the point group of its lattice is the number of orientations available; 25 of the thirty-two have more than one, and the 7 holohedral classes have exactly one — their crystal already has every symmetry their lattice has, so there is nothing left over to twin by. The names along the right are the old mineralogical ones: hemihedral for half, tetartohedral for a quarter.

Symmetry at work

Twenty-five of the thirty-two can twin, and seven cannot

The number of twin laws available to a crystal is the index of its class in the point group of its lattice, minus one. Doing that arithmetic for all thirty-two classes takes a moment and produces a census with a sharp edge on it — the seven classes that cannot twin this way are exactly the seven that already use everything their lattice has.

p4 inside p4m, by area. A fundamental domain for p4m beside one for p4, drawn by the same construction on the same grid. p4 sits inside p4m with index 2: it has 8 ÷ 4 = 2 times as few operations to rebuild the pattern with, so it needs 2 times as much of the cell to rebuild it from — measured here at 1.51 times as many samples. The two domains are one region and 2 copies of it, and the ratio between them is the index rather than a coincidence of shape.

Operations

Domains of a subgroup

A group with half the operations needs twice as much of the cell to rebuild the pattern from. That single sentence is the index arithmetic of the whole classification, and it turns the containments among the seventeen into a statement about area.

$The eleven screw axes, enumerated. An n-fold axis admits a screw for every m from 1 to n − 1, so the orders a lattice permits — 2, 3, 4, 6 — give 11 screws in all. Each row shows the fraction of a cell one turn advances, plotted between zero and one. 3 of them are their own mirror image, which happens exactly when m is half of n; the others pair off into left- and right-handed twins.$ Into space

Eleven ways to turn while climbing

Five rotation orders survive the crystallographic restriction, and each of them admits a screw for every whole number of cells its turns can amount to. That is a sum with four terms, and it is why there are eleven screw axes rather than some other number.

The icosahedral group, counted. The sixty rotations of an icosahedron, found by trying every map that sends one adjacent pair of vertices to another and keeping those that carry the whole vertex set onto itself. They fall into 6 axes of order 5, 10 axes of order 3, 15 axes of order 2 — and the fivefold axes are the reason this group cannot be the point group of any crystal, since no three-dimensional lattice admits a rotation of order five. Quasicrystals have it anyway, which is what made 1982 an argument rather than a measurement.

Order without repetition

Icosahedral symmetry

Sixty rotations, six fivefold axes, and no lattice in three dimensions that can hold any of them. It is the point group a crystal is forbidden, and the one the first quasicrystal turned out to have.

A fundamental domain for p6m. One representative from every orbit of p6m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

The classification

Orbifold notation, the shorter language

Fold a pattern up along its own symmetries and what remains is a small surface with marked points. Its shape is a complete name for the group, and reading the name off costs an arithmetic sum that has to come to two.

$The eleven Laue classes. Adjoining the inversion to each of the thirty-two crystal classes collapses them onto 11 groups. Friedel's law says a diffraction experiment sees the crystal and its inverse alike, so this — and not the crystal class — is what a diffraction pattern's symmetry reports. The highlighted symbol in each row is the class that is already its own Laue class, which is to say the centrosymmetric one.$ What symmetry decides

The eleven a diffraction pattern reports

A diffraction experiment cannot tell a crystal from its inverse. So the thirty-two classes collapse to eleven before a single reflection is indexed, and a structure determination begins by answering a different question from the one it was asked.

The twin laws of class 32. The point group of the lattice of class 32 has 24 operations and the class has 6, so it splits into 4 cosets: the crystal itself, and 3 twin laws. Every operation in a block produces the identical second orientation, which is why the block and not the operation is the law.

Symmetry at work

Quartz has exactly three twin laws, and its lattice is why

Class 32 on a hexagonal lattice has index four, so three twin laws and no more. They turn out to be the three the mineralogists named — Dauphiné, Brazil and the combination of the two — and reading quartz's lattice off its class instead of measuring it would have produced one law where there are three.

The h0l layer of P2₁/c. The h0l reflections of P2₁/c out to 4 in each index, with each spot decided by summing the structure factor over the group's operations: 44 survive and 36 vanish identically, whatever the atoms are. The pattern of holes is the condition h0l: l even, read back off the spots rather than imposed on them.

How it is known

The reflections that are not there

A screw axis and a glide plane leave no mark on the intensity of any reflection. What they do is delete some, exactly, for every possible arrangement of atoms — and the pattern of deletions is computed here from the sum a crystallographer writes down, rather than read from a table.

Neumann's principle for elastic constants in mmm. Each bar is one operation's contribution to the character of the representation the elastic constants live in — the stiffness of the crystal in every direction and shear. The identity contributes the unconstrained count of 21; every other operation of mmm subtracts from it, and the average over all 8 is 9, which is how many independent components the property may have. The sum is exact in integers and is taken in the lattice basis, so no Cartesian frame is chosen anywhere in it.

What symmetry decides

Neumann's principle, as one sum

A physical property of a crystal must be unchanged by every symmetry the crystal has. That is a whole subject in one sentence, and it reduces to arithmetic: how many independent components a property may have is a character averaged over the point group, exact in integers.

The five kinds of glide plane. All five glide letters: a, sliding by a/2; b, sliding by b/2; c, sliding by c/2; n, sliding by (a+b)/2, (b+c)/2 or (a+c)/2; d, sliding by (a+b)/4, (b+c)/4 or (a+c)/4. 3 of those drawn are axial — the slide is half of a single cell vector lying in the plane. A glide's square is a pure translation, so its slide can only be half of a lattice vector, and the quarter-cell d glide exists only where centring has already made the half-diagonal a lattice vector.

Into space

Reflect, then slide by half of something

A glide's slide must double to a lattice vector, which leaves three candidates in any plane and a fourth that exists only where centring has already made a half-diagonal into a lattice vector. Five letters, and the fifth is the one the enumeration explains.

How often a dot gives the wrong group. Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it.

The classification

The groups ornament actually uses

Seventeen exist and decoration does not use them evenly. Which are common is an empirical question that published surveys answer differently, and part of the reason is a hazard this site can measure exactly.

$A chain modulated at q = 0.211. The lower row is the lattice: 34 sites, evenly spaced. The upper row is the structure: the same sites displaced by a wave of amplitude 0.12 of a spacing and wavevector 0.211, drawn through them. Because 0.211 is not a ratio of small whole numbers, no cell of any size holds the structure — the displacement pattern never repeats — and yet the atoms are nowhere near random: each one is exactly where a single sine wave says it should be. That is what an incommensurately modulated crystal is, and its diffraction pattern is sharp.$ Order without repetition

The satellites that need a second integer

A crystal whose atoms are displaced by a wave of the wrong wavelength has no unit cell at all, and diffracts to sharp spots anyway. Indexing them takes two integers per reflection instead of one — and the intensities of the extra spots are Bessel functions, which is a check the arithmetic can be made to pass.

$A structure, and the vectors between its atoms. On the left, 4 atoms in a cell. On the right, every one of the 16 vectors between them, each drawn from a common origin: 13 distinct positions, with the 4-fold peak at the origin being each atom paired with itself. That right-hand picture is what a Patterson map shows, and it is the thing a diffraction experiment gives without phases. It has more peaks than the structure has atoms — n² against n — which is why interpreting one is hard, and why it is always symmetric about its centre.$ How it is known

The map that needs no phases

A diffraction experiment measures intensities and loses phases, so the electron density cannot be computed from it. One map can be: the transform of the intensities, whose peaks are not atoms but the vectors between them — every ordered pair, brought to a common origin.

Lattices

Twenty-five cells, and fourteen lattices

The usual picture of the fourteen Bravais lattices is a plate of fourteen boxes, which is the answer with the argument removed. The argument is one question asked of twenty-five candidates, and the question has a computable answer.

m3̅m → 4mm: 6 domain states. The transition from class m3̅m to class 4mm loses 40 of the parent's 48 operations, so the child has index 6 and the crystal comes apart into 6 domain states. Each colour is one state — one coset of 4mm in m3̅m — and each holds the same 8 poles. The lost operations are what carries one state onto another, and they survive in the crystal as the relation between its domains rather than as symmetries of any part of it. The descent changes the crystal system, so the states differ in shape as well as in orientation and the transition is ferroelastic.

Symmetry at work

How many domains a transition makes is an index

Cool a crystal through a symmetry-lowering transition and it has to choose one of several equally good low-symmetry arrangements. Different parts of it choose differently, and the number of available choices is the index of the new group in the old one — a number available before any crystal is grown, and one of the few predictions in this field that is a count rather than a bound.

Why seven — all 16 candidates. Every subset of the 4 extras available on a strip, closed under composition and named from the operations that come out. 16 candidates give 7 distinct groups: 9 of them generate operations they were not given and land on a group already listed.

The classification

Why sixteen become seven

Four extra operations give sixteen combinations and seven groups. The nine that vanish are not cases anybody forgot — each one comes back from the closure holding something it was never given, and one of them changes the lattice underneath it.

P2₁/c under every cell choice. One group, described in each of the 6 bases that keep its cell the shape its system requires, with the symbol derived from the operations each time. The distinct symbols are P2₁/c, P2₁/a, P2₁/n — 3 names for one group. Nothing about the crystal has changed: the basis changes all have determinant ±1, so the lattice is untouched, and the census of rotations, screws, mirrors and glides is identical in every row, since conjugation cannot turn one kind into another. What changes is which lattice vector a glide's translation is half of, and the glide letter names exactly that.

Into space

One group, three symbols

P2₁/c, P2₁/a and P2₁/n are the same space group written on three choices of axes, and the literature contains all three as though they were different. Deriving a symbol from a group's own operations shows why — and found an entry on this site that had been carried under another setting's name for two phases.

Everything class m3̅m can descend to. The 25 crystal classes that are subgroups of m3̅m, arranged by order, with the 56 maximal steps between them drawn as edges. The order of each row is printed down the left, so the index of any step is the ratio of the two rows it joins. A symmetry-lowering transition can only be continuous when it goes down one of these edges, and the index on the edge is the number of domain states the transition produces. A descent of several steps is possible but has to happen discontinuously or through the intermediate classes.

Symmetry at work

The descent of symmetry is a lattice, not a tree

Which classes a crystal can fall to when it loses symmetry, drawn as a graph with the index on every edge. It is routinely called a tree and it is not one — a class can be reached from its parent by several different routes of the same total index, and which route a material takes is a physical question the diagram deliberately leaves open.

The classification

The friezes inside the seventeen

Take one lattice row of a wallpaper pattern and keep only the symmetries that leave that row where it is. What survives is a frieze group — and which of the seven it turns out to be is a fact about the plane group that its symbol does not state.

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