Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: The lattice forbids

Periodicity is a strong constraint. It rules out five-fold rotations, most rotation orders above six, and a great many patterns that look perfectly reasonable until the arithmetic is done.
A Penrose tiling, 4 inflations. Two rhombs, subdivided into smaller copies of themselves over and over. The result covers the plane, has five-fold symmetry about its centre, and never repeats — there is no translation that maps it to itself. Order without repetition

Penrose tilings

Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice.

 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 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.

 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.

 A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call. Order without repetition

Order is not periodicity

For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.

 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 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.

 One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does. Lattices

Centring, and why cm is not pm

A centred cell has a lattice point in the middle and twice the area it needs, and crystallography prefers it anyway. The preference has a price, and the price is paid in reflections that vanish for reasons that have nothing to do with the crystal.

 An integer matrix of order 5. The companion matrix of the 5th cyclotomic polynomial has whole-number entries and order exactly 5, so it is a genuine 5-fold symmetry of a 4-dimensional lattice. The plane it rotates sits at an irrational angle to that lattice, and the lattice's shadow on it is dense — which is why a projection needs a window before it becomes a pattern. What a lattice forbids

Where five-fold becomes legal

A five-fold rotation with whole-number entries exists — in four dimensions, as a four-by-four matrix that can be written down. The plane forbids it because the plane is too small, and knowing which dimension is large enough changes what a quasicrystal is.

 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.

 Cut and project. A square lattice, a strip along a line of the given slope, and the shadow on that line of every lattice point inside the strip. The shadow has two gap lengths; whether their order repeats depends entirely on whether the slope is rational. Order without repetition

Cut and project

Take a periodic lattice, cut a strip through it at an irrational angle, and keep the shadow of what falls inside. The result never repeats, has exactly two spacings, and is a quasicrystal — built from something perfectly periodic that is simply not where anybody was looking.

 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.

 A 5-fold cluster in a crystal that has no 5-fold axis. A cluster of 10 points with an exact 5-fold axis at the centre of each cell, repeated by the lattice. Two measurements, on the same points. The cluster is carried onto itself by a turn of 72° to within 2e-16 of a cell — exact, as far as the arithmetic goes. The pattern is not: applying the same turn about a lattice point sends some atoms 1.19 of a cell from the nearest atom, which is most of the way across it. Both are true at once. The axis is a symmetry of the contents of one cell and not of the crystal, which is what non-crystallographic symmetry means and why a virus with a sixty-fold capsid can crystallise in an ordinary space group. What a lattice forbids

A fivefold axis in an ordinary crystal

A virus with sixty-fold symmetry crystallises in a space group that has none of it. The restriction forbids a fivefold axis to the lattice and says nothing about what sits inside one cell — so the axis is exact, the crystal genuinely lacks it, and both statements are measurable on the same set of atoms.

 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.

 A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call. Order without repetition

What Shechtman measured

A diffraction pattern with sharp spots and tenfold symmetry, in April 1982. Sharpness meant order and tenfold meant no lattice, and the two had been believed inseparable — so the interesting question is what the observation had to rule out before it could mean anything.

 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.

 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.

 4 lattices. 4 lattices: cubic P, with 48 symmetries; cubic C, with 16 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything. Lattices

Forty-eight becomes sixteen

Centre one face of a cube and the four threefold axes along its body diagonals are gone. That sentence is usually offered as a fact to accept; it is a computation whose answer is a number, and the number says which lattice you got instead.

 How many ways each class can lose symmetry. Every crystal class, with the number of distinct classes it can descend to — 247 parent-and-child pairs in all across the thirty-two, counted up to conjugacy in the parent, which is the equivalence that says two descents differing only by which axis was chosen are one transition. The count rises steeply with the order of the parent, which is why the cubic and hexagonal holohedries dominate the list of materials with rich domain structures. Symmetry at work

Two hundred and forty-seven descents, or two hundred and twelve

How many distinct ways can a crystal lose symmetry? Counting parent-and-child pairs up to conjugacy in the parent gives 247. The standard enumeration in the ferroics literature gives 212, and the operation that merges the extra thirty-five turns out to be a rotation through forty-five degrees — which no lattice may have, and which no integer matrix in a lattice basis can therefore express.

 Which indices have a square sublattice. For each index up to 26: how many sublattices of the square lattice are themselves square, found by testing whether the quarter-turn maps each one onto itself; the same count as a sum over divisors, +1 for each divisor one more than a multiple of four and −1 for each one less; and the ways of writing the index as a sum of two squares. The three agree at every row, which is Fermat's theorem — and it says that 3, 7 and 11 have no square sublattice at all while 5, 13 and 17 have two. Lattices

The sublattices that stay square

A sublattice of the square lattice is itself square exactly when its index is a sum of two squares — so index five has two and index seven has none, and which superstructures a surface can form is decided by a theorem of Fermat's about primes.

 Every coincidence index is odd. The rotations that bring a cubic lattice into coincidence with itself, up to index 25: 17 distinct relations across 12 indices, each found by enumerating integer quaternions and each index computed as the size of a sublattice rather than from the usual formula — the two are then required to agree. 5 of the 12 indices carry more than one relation, which is why the tables write 13a and 13b. Every index is odd. That is not a feature of this range: a rational orthogonal matrix written in lowest terms has an odd denominator, so the factors of two always cancel. Symmetry at work

Every coincidence index is odd, and in the plane most of them do not exist

The indices at which two copies of a cubic lattice share points are 3, 5, 7, 9, 11 and every odd number after them. There is a two-line proof that no even index can occur. Ask the same question about a square lattice and the answer is a different list entirely, governed by which numbers are sums of two squares — so Σ3, which is the commonest boundary in every metal, has no plane analogue at all.

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