Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Exactly this many — page 10

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'.
The sphere fixes a count; the torus fixes only a difference. Euler's relation for a trivalent net gives Σ (6 − n) pₙ = 6χ, so the surface fixes one linear combination of the face counts and nothing else. On a sphere that combination is twelve, which with no face smaller than a pentagon forces exactly twelve pentagons. On a torus it is zero, which permits any number of pentagons provided as many heptagons pay for them — and permits none at all, which is the plain hexagonal net. On a surface of two holes it is minus twelve, so heptagons become compulsory instead. What a lattice forbids

As many heptagons as pentagons

A trivalent net on a sphere must have exactly twelve pentagons. The same three lines of arithmetic on a torus give zero — which does not forbid pentagons, it makes them pay: every pentagon has to be balanced by a heptagon, and the counts are otherwise free. One rotated bond in a wrapped honeycomb makes two of each and changes nothing else.

 No two cubic grains are more than sixty-three degrees apart. For each proper class: how many rotations describe one misorientation, the largest disorientation there is, and the mean over uniformly random orientations. The maximum is found by sampling and then climbing locally, so it is a lower bound that has stopped moving rather than a solved value — and it lands on the numbers the literature records. Symmetry at work

The angle two grains differ by

A crystal's axes are not labelled, so a relative orientation between two grains has as many descriptions as the symmetry allows — five hundred and seventy-six of them for a cubic crystal — and their rotation angles run from a few degrees to more than a hundred and seventy. The honest answer is the smallest, and its largest possible value is a number: no two cubic grains are more than sixty-three degrees apart, whatever anybody does to them.

 The parity argument loses 36 pairs it had won alone. The argument that refutes ten of the twenty-one species walks round a polygon of odd size: the ring of polygons about it is a closed walk of odd length in a graph the species decides, and a bipartite graph has no such walk. With two species at a vertex the flanking pairs come from the union of two graphs, and a union of bipartite graphs need not be bipartite — so the walk stops being constrained. The fourth row is the cost: pairs whose members the argument kills on their own and which it cannot kill together. The classification

The argument that closes eleven

Twenty-one vertex species satisfy the angle equation; a parity argument kills ten before anything is drawn, and the eleven survivors are all built. Asking the same question of tilings with two kinds of vertex, the parity argument evaporates — it constrains a walk in a graph one species decides, and two species decide the union of two graphs, which need not be bipartite. What is left is a search, and a search cannot close a count.

 Four angles, and the integer that picks them. Two roots at angle θ have Cartan integers whose product is 4cos²θ. Both are whole numbers and the product is below four, so it is nought, one, two or three — and each value fixes the angle between the two roots, and with it the angle between the mirrors perpendicular to them. The shaded wedge is the region the pair of mirrors folds the plane onto; the smaller it is, the larger the group they generate. What a lattice forbids

Four root systems, and the same four rotations

Two mirrors meeting at an angle generate a group. Ask that the group be finite and that a certain pairing between the mirrors come out a whole number, and the angle has only four possible values — from which the rotations that survive are of order two, three, four and six. The crystallographic restriction arrives with no lattice anywhere in the argument.

 The cross, from the selection rule alone. The layer lines of a helix with the first maximum of each marked on both sides. Nothing here is a picture of a photograph: each mark is at the radius where the Bessel function of the lowest order the selection rule permits on that layer line first peaks, and that radius is proportional to the order. The order rises by one per layer line until the middle of the repeat, so the maxima lie on two straight lines through the origin — the X — and the larger marks are the layer lines that reach the axis. The classification

What a thread scatters

A helix with ten subunits in a turn is not a screw axis a crystal may have, and nothing about its diffraction pattern is lawless. The pattern lies on layer lines, and on each one only certain angular orders may contribute — a selection rule as hard as any extinction condition. The lowest permitted order rises by one per layer line, a Bessel function of order n does nothing until its argument is about n, and the maxima therefore lie on two straight lines through the origin.

 One change of basis turns a Gram into its own adjugate. For each Gram matrix: the matrix after the basis change by a right-angle rotation, and the adjugate. They are equal, always — and the adjugate is the determinant times the inverse, which is the dual lattice's Gram. So the dual is the same lattice on a rotated basis, scaled by one over the determinant. Five rows are the named plane lattice types and the rest have entries picked at random, because the claim is an identity in integers and not a property of the five. Lattices

Every plane lattice is its own dual

The dual of a lattice has the inverse Gram matrix, and in two dimensions the inverse is the adjugate over the determinant — which is what one particular change of basis does to a Gram. So a plane lattice's dual is the lattice itself, turned through a right angle and scaled, for every lattice with no exception. In three dimensions it is a condition, and the face-centred and body-centred cubic lattices are duals of each other rather than of themselves.

 One group refuses two colours and three refuse three. The two counts side by side, with the rows that refuse a number of colours marked. p3 is the only group with no two-colouring; p4, p4m and p4g are the only ones with no three-colouring. Neither list is a subset of the other and both come from the same arithmetic — a rotation order that divides nothing the symmetric group has. The classification

What a half-turn does to three colours

Ten of the seventeen plane groups have no three-colouring, because a half-turn cannot permute three colours cyclically — that is the first rung of this ladder and it is true. Drop the word cyclically and the answer changes completely: a half-turn permutes three colours perfectly well by swapping two and fixing one, and only the three four-fold groups refuse three colours at all.

 12 of the thirty-two classes have a free invariant ring. Every crystal class with its order, the number of its operations that are reflections, whether its ring of invariant polynomials is free, and the degrees of the generators when it is. A reflection here is an operation of determinant minus one whose fixed set is a plane; an inversion centre has determinant minus one and fixes only the origin and is not one. The classes with a free ring are exactly the classes generated by their reflections, which is Chevalley's theorem checked rather than quoted. What symmetry decides

Twelve of the thirty-two are free

A crystal class leaves some polynomials alone, and the ones it leaves alone form a ring. For twelve of the thirty-two classes that ring is generated by three polynomials with no relation between them, and for the other twenty it is not — and the twelve are exactly the classes generated by their mirror planes. The two verdicts are computed by routes sharing no code, and an inversion centre is not a mirror.

 Perfection is a rank, and most lattices do not reach it. For each lattice, the rank of the matrices vvᵀ built from its shortest vectors, against the dimension of the space of symmetric matrices those live in. Reaching it means the shortest vectors pin the form down completely: no deformation keeps every one of them at its length. Falling short means there is a direction left to move in, and the lattice is not a local maximum of density. Lattices

One perfect form in space

Which lattice packs spheres most densely is a question about a maximum over a continuum, and Voronoi turned it into a rank calculation and a sign check. A lattice is a local maximum exactly when its shortest vectors pin its shape down completely and its inverse can be written over them with positive coefficients. Searching every reduced integer form of minimum two finds one such lattice in the plane and one in space.

 A spin needs two full turns to come back. The number a rotation about a fixed axis multiplies a state by, against the angle turned through. A vector — anything of integer spin — is back where it started after one full turn; a spin-one-half state is multiplied by minus one and needs a second turn. So the operators acting on such a state do not form the rotation group: a full turn is an operation distinct from doing nothing, and the group is twice as large. What symmetry decides

Two turns to come back

A rotation through a full turn does nothing to a crystal and multiplies a spin-one-half state by minus one, so the group acting on such a state is not the point group but a group twice its size. Building those eleven double groups from quaternions and averaging a random operator over each gives the degeneracies a spin may have — and shows that the doubling everybody calls Kramers' is time reversal's doing and not the double group's.

 How many similar sublattices the cubic lattice has at each scale. Every integer matrix satisfying MᵀM = α²I, counted up to the lattice's own point group by marking orbits rather than dividing. The even scales are drawn apart because they are the ones that give nothing new: a factor of two in the scale never produces a shape the smaller scale did not already have. Lattices

The shapes a lattice in space can thin to

In the plane, which indices admit a sublattice of the same shape is a question about which integers a quadratic form represents, and Fermat answered it. In space the question collapses: taking determinants shows the index is always a perfect cube, so there is nothing to represent. What is left is how many there are at each cube — and for a hexagonal lattice, whether there are any at all depends on one number.

 Thirty-two classes, from fourteen Gram matrices. The five hundred and ten subgroups sorted by how many operations of each kind they contain — a determinant and a trace decide which of the ten kinds a matrix is. Thirty-two answers come out, and they are the thirty-two crystal classes: matched against the construction elsewhere in this collection by signature rather than by name, since nothing here names a point group. What symmetry decides

Thirty-two from fourteen matrices

Write the fourteen Bravais lattices as Gram matrices, ask each one which integer matrices preserve it, and take every subgroup of every answer: five hundred and ten of them. Sort those by how many operations of each kind they contain — which a determinant and a trace decide — and thirty-two answers come out. They are the crystal classes, from a construction in which no point group is ever named.

 Four vectors summing to zero, and six numbers on the edges. A superbasis is the three basis vectors together with their negated sum, so the four sum to nothing and their pairwise products sit on the six edges of a tetrahedron. Selling's rule is: while any edge is positive, apply one transformation. The right panel is the same lattice reduced, with the vanishing parameters marked — and a vanishing parameter is a face the Voronoi cell does not have. Lattices

A reduction with one rule

Niggli's reduction is eight numbered conditions with sub-cases, applied in order until none applies. Selling's is a single rule on four vectors that sum to zero: while any of six numbers is positive, do one thing. It terminates sooner, its termination is a quantity that visibly falls, and when it stops the six numbers are the Voronoi cell — the pattern of which ones vanish gives Fedorov's five solids and nothing else.

 Twenty-two halvings the fourteen lattices permit. Every lattice has exactly seven subgroups of index two, whatever its shape. The third column is how many of the seven the lattice's own group carries onto themselves, and the fourth is how many of those survive as distinct types once a change of basis within the type is allowed to identify them. The running total ends at twenty-two, which with the fourteen grey lattices is the thirty-six magnetic Bravais lattices — and the row that ends at zero is the face-centred cubic lattice. Lattices

The halving a lattice will not permit

Admit time reversal and a lattice splits into points that leave the moments alone and points that reverse them. The second set is a coset of a subgroup of index two, and every lattice has exactly seven of those, whatever its shape. What differs is how many of the seven the lattice's own symmetry survives — and the face-centred cubic lattice survives none of them.

 The kinds of line defect each breaking allows. The eleven proper crystal classes, each with the order of the binary group that covers it, the number of conjugacy classes of that group other than the identity — which is the number of kinds of line defect — and whether the group commutes. 6 of the eleven do not, and in those media two defect lines cannot pass through each other without leaving a third line behind. Symmetry at work

The defect that needs two laps

Which defects a medium can have is not a fact about the medium. It is a fact about the space its order parameter lives in, and for a rotational symmetry broken down to a point group that space has a fundamental group twice the size of the point group. The kinds of line defect are its conjugacy classes — and in six of the eleven cases they do not commute, which means two defect lines cannot pass through each other.

 Which indices a screw axis contains itself at. The fifteen kinds of axis a space group may have, against the index of the sublattice taken along the axis. A filled cell is an index at which the axis contains a copy of its own kind; the darker cells are the indices at which what comes back is the mirror image instead. A pure rotation axis is filled everywhere and a screw is not, and which indices a screw loses is decided by one congruence rather than by any geometry. Into space

A screw that contains its own mirror image

No operation of a crystal turns a right-handed screw axis into a left-handed one — that is what makes the eleven enantiomorphic pairs pairs. And yet a 4₁ axis contains copies of 4₃ as subgroups, at every index congruent to three modulo four. One congruence decides both which indices are possible and which hand comes back, and it is the same congruence for all fifteen kinds of axis.

 Seventy-three arithmetic classes, from fourteen groups. Every subgroup of every lattice's own group, split by whether the subgroup's own Bravais group is that lattice's. The ones that are not belong to a lower lattice and are counted there, which is what stops the same class being counted twice. The running total ends at seventy-three, and no conjugacy in GL(3, ℤ) was ever decided. What symmetry decides

Seventy-three, without a search

The unit a space group is built from is a point group together with the lattice it acts on, and there are seventy-three of them. Getting there looks like it needs conjugacy in GL(3,ℤ), which is a search this collection tried and abandoned. It does not: every finite group of integer matrices carries a canonical larger group that says which lattice it belongs to, and once that is computed the search has nothing left to do.

 Where the laminate's middle eigenvalue crosses zero. The middle eigenvalue of FᵀF − I for the average deformation of a twinned laminate, against the volume fraction of one variant. At both ends the laminate is a single variant and the value is well away from zero; in between it crosses, twice, and each crossing is a volume fraction at which the laminate can meet the parent phase across a plane. The two roots are complementary, which is the same plate with the two variants exchanged. Symmetry at work

The plate that only fits when it is twinned

No single variant of a cubic-to-tetragonal transition can meet its parent across a plane. A fine mixture of two variants can, because its average deformation carries a free parameter — the volume fraction — and that parameter passes through the compatibility condition twice. Sweeping it gives the two fractions, the two habit planes, and one inequality: the plate exists exactly when the two principal stretches satisfy η₁² + η₃² ≤ 2.

 Why a tetrahedron is not a cube cut up. The two invariants side by side. A cube's twelve right angles are each a rational part of a turn and contribute nothing; a regular tetrahedron's six edges each contribute one α, giving six. Cutting a polyhedron and rearranging the pieces cannot change the invariant, so no dissection takes one to the other however the volumes are matched. That is Hilbert's third problem, and the whole of it is one angle. The classification

The angle that is not a fraction of a turn

Any two polygons of equal area can be cut into pieces that rearrange into each other. In space that fails, and the obstruction is a sum over edges of length against dihedral angle — zero for anything that fills space, and not zero for a regular tetrahedron. The whole argument reduces to one claim about one angle, and that claim is an integer computation: a sequence that is never divisible by three, when it would have to be.

 Three places to put the boundary. A large site and a small one, with three candidate boundaries between them. Halfway is the ordinary Voronoi cell and it cuts through the large sphere. Splitting in the ratio of the radii is the natural repair and its surfaces are not planes, so the cells do not fit together. The power plane sits where the tangent lengths agree, which is further from the large site than halfway and is still a plane — and being a plane is the whole reason the construction works. Lattices

Where the boundary goes when the atoms differ

Assigning each point of space to the nearest atom is the right rule only when every atom is the same size. Splitting the distance in the ratio of the radii is the obvious repair and it produces curved faces that do not fit together. The repair that works measures to a sphere rather than to a point: the boundary stays a plane, the cells still tile exactly, and a small enough atom loses its cell altogether — at a radius ratio of exactly one in eight.

 Cubic means the Sylow 3-subgroup is not normal. The thirty-two sorted two ways at once: by crystal system and by whether the Sylow 3-subgroup is normal. Two of the four boxes are empty, so the two properties coincide exactly. That gives 'cubic' a definition with no geometry in it — a class is cubic when its threefold subgroups are conjugate to each other rather than unique — and it explains why a cubic class has no principal axis: a group cannot single out one member of a conjugate family. What symmetry decides

How many axes there are is a Sylow count

Sylow's theorems say that the subgroups of prime-power order in a finite group are all conjugate and that how many there are is congruent to one modulo the prime. Applied to the thirty-two crystal classes that arithmetic counts axes: the number of Sylow 3-subgroups is the number of equivalent threefold directions, it is four in exactly five classes, and those five are the cubic ones. So cubic has a definition with no geometry in it.

 Three lattices no congruence can separate. The three reduced forms of discriminant minus twenty-three, with the integers each represents. The principal form represents one and the others do not; the others represent two and it does not. So they are genuinely different lattices — and they represent exactly the same residues modulo twenty-three, so they are in one genus and no congruence condition of any kind distinguishes them. Lattices

Lattices that agree at every prime

Counting the plane lattices with a given metric determinant is a class number. Above it sits a coarser count — the genus, which is what congruences can see — and for most small determinants the two agree. At discriminant minus twenty-three they part: three lattices representing exactly the same residues modulo everything, and different integers. No argument modulo any number can tell them apart, and they are not the same lattice.

 The five, as generator counts. Each of the five convex bodies that tile space by translation, built as the set of combinations of a handful of vectors with coefficients between zero and one. Three generators give a cube, four give either a hexagonal prism or a rhombic dodecahedron depending on whether three of them are coplanar, five give the elongated dodecahedron and six the truncated octahedron. The last column is what the same number of generators would give in general position, and the shortfall is the number of faces lost to coplanarity. The classification

Every parallelohedron is a shadow of a cube

Take a few vectors and form every combination of them with coefficients between zero and one. All five of the convex bodies that tile space by translation come out of that recipe, from three vectors, four, four, five and six — and since the recipe is exactly the image of a cube of that many dimensions, the truncated octahedron is a three-dimensional shadow of a six-dimensional cube. The five are not the generic answers: they are the degenerate ones, and the degeneracy is what the tiling demands.

 The three minima of seven lattices. Every lattice scaled to covolume one, with the smallest radius at which a ball holds one, two and three independent lattice vectors. The last column is the shortest vector as a fraction of the longest any lattice of this volume can have — Hermite's constant — and only the face-centred cubic lattice reaches it. The fifth column is the product of the three, which is capped whatever the lattice. Lattices

A lattice cannot have all its vectors long

The three successive minima are the radii at which a ball first holds one, two and three independent lattice vectors. Nothing bounds any of them above on its own — a cell can be flattened without limit — but Minkowski's second theorem caps their product, so pushing one up forces another down. That is why every crystal has a shortest direction worth naming, and why a very anisotropic cell has a very short one.

All themes · All essays