Crystallography & Symmetry
patterns derived from their groups
Concept

Higher-dimensional lattice — where it appears

A periodic lattice in more than three dimensions, used to describe an aperiodic structure as a slice of a periodic one. The extra dimensions are bookkeeping rather than a claim about where the material is, and they are what make the arithmetic decidable.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

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.

restriction · Restriction 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.

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.

aperiodic · Quasicrystals A quasilattice down a 5-fold axis: 10-fold, from an axis of order 5. 153 points of a three-dimensional quasilattice, made by keeping the points of Z⁶ whose perpendicular image lies inside a window and projecting them into ordinary space, then viewed along one of its 5-fold axes. There is no lattice here and no unit cell, and the symmetry is nevertheless exact: all sixty rotations of the icosahedral group carry the set onto itself, on 153 points of its core, measured by applying them rather than assumed from the construction. Seen down this axis the set comes back to itself under a turn of a 10th and no finer turn, measured over every turn up to a twelfth — twice the order of the axis, and the factor of two is a centre of symmetry: this set equals its own negation, which is what a window centred on the origin produces, and that is checked here rather than assumed. A diffraction experiment would record the same 10 whatever: the measured intensity acquires a centre whether or not the structure has one, so the tenfold photograph of 1982 does not by itself distinguish a structure with a centre from one without.

Six integers, and the lattice that holds them

A fivefold rotation is not an integer matrix in three dimensions and is one in six. The icosahedral group permutes its own six fivefold axes, so in coordinates along those axes every one of its sixty rotations is a signed permutation — and Z⁶ is a lattice it maps onto itself.

aperiodic · Quasicrystals In the plane, the lengths do name the lattice. Every reduced binary form with coefficients up to 20 — 1750 lattices — with its theta series computed to 120 terms. No two of them agree. That is Schiemann's theorem for binary forms, which says the theta series determines the lattice in two dimensions and in three, confirmed here as far as the search reaches rather than proved. The closest pair is worth the space: two lattices whose shortest vectors both have squared length twenty agree for 38 terms — because neither has any vector before then — and part at the next one.

The lengths do not name the lattice

Seventeen hundred plane lattices, every one with a theta series shared with no other — the lengths determine the lattice, and an exhaustive search says so. In sixteen dimensions two different lattices have identical counts at every distance, and the example is sixty years old.

lattices · Lengths 66 squares and 106 rhombs. The Ammann–Beenker tiling, built by keeping the points of a four-dimensional lattice whose companion image falls inside an octagon and projecting them into the plane. Every tile has the same edge length; the squares and the forty-five degree rhombs are told apart by their diagonals. Nothing was placed — the faces were found among the projected points.

Eight-fold, with the golden ratio taken out

The familiar quasicrystals are all built on five: Penrose's rhombs, the Fibonacci chain, the ten-fold pattern Shechtman measured. A method that works only on the golden ratio is a method tuned to its answer — so here is the same construction run on eight, where the irrational is √2 and nothing else changes.

aperiodic · Quasicrystals Two, seventeen, two hundred and thirty, and then. The number of arithmetic crystal classes and the number of crystallographic groups in each of the first six dimensions, with the second divided by the first. The classes multiply by between five and fourteen a dimension; the groups multiply by much more, and the quotient — how many groups an average class carries — goes 1.00, 1.31, 3.15, 6.74, 36.5 and 339. The last column says what is derived on this page and what is quoted: the plane in full, six of the seventy-three classes in space, and nothing at all above three dimensions, where the counts come from machine enumerations of the 1970s onwards.

Finitely many is not few

Bieberbach's third theorem says each dimension holds finitely many crystallographic groups and gives no idea how many. The counts are 2, 17, 230, 4783, 222018 and 28927922, and dividing them by the number of arithmetic classes says which of the classification's three steps supplies the explosion — the step that attaches translations, not the one that finds the matrix groups.

restriction · Finiteness Three rotations of order twelve, and one twelve-fold. Three integer 4 × 4 matrices of order twelve, one from each rational class of that order on a four-dimensional lattice. A four-dimensional rotation turns two perpendicular planes, and each plane is drawn as a dial: a point, its images, and the path the operation takes round them. Φ₁₂ turns its planes by one twelfth and five twelfths of a turn, so a point runs round twelve positions. Φ₃ ⊕ Φ₄ turns one plane by a third and the other by a quarter, and Φ₄ ⊕ Φ₆ by a quarter and a sixth; each has order twelve because twelve is the least common multiple, and neither has a plane with twelve positions in it.

Three rotations of order twelve

A four-dimensional lattice can be carried onto itself by three different integer matrices of order twelve, and only one of them turns anything by a twelfth of a turn. The totient that is usually said to decide which orders a dimension permits decides something else: which angles. The orders are decided by a sum over blocks, the two lists part company at six dimensions, and space already has an operation whose order is not its angle.

restriction · Restriction One group without an axis, and as many as the order with one. For a rotation of each order that an integer matrix can have in a small dimension, the number of space groups its arithmetic class admits — computed from the cohomology rather than enumerated. A rotation acting on the smallest lattice that will hold it fixes no direction and admits exactly one group: the symmorphic one, with no screw. Add a direction it leaves alone and the count becomes the order of the rotation, and the extra groups are its screws. The four-fold with an axis gives four, which are P4, P4₁, P4₂ and P4₃; the five-fold with an axis gives five, in five dimensions, where no published table exists to check it against.

The screw a dimension does not have

The extension count is a machine that runs in any dimension, and the seventeen were the case where every step could be checked against a list arrived at four other ways. Run on a cyclic point group it has a closed form two lines long — and it says a five-fold screw axis does not exist in four dimensions, which is a prediction rather than a check.

classification · Cohomology The centrings six four-dimensional lattice groups allow, prime by prime. For each of the six four-dimensional lattices whose symmetry groups are maximal, with the group's order, and each prime from 2 to 7: the subspaces of the lattice modulo that prime the group keeps, sorted by what centring by each one makes. New Bravais types appear only at 2, three of them, all over the cube-and-line lattice, and at 5, where A₄ centred by fifths is A₄*. Both centrings of the hypercubic lattice by halves make D₄, which has three times the symmetry.

The body-centred hypercube is the checkerboard lattice

In three dimensions the centring letters come out as the subspaces a holohedry keeps modulo a prime, and only two and three make new lattices. In four dimensions the same census, run on the six lattices whose symmetry groups are maximal, finds a prime that the three-dimensional argument never could: five centres A₄ into its dual. It also finds that the most familiar centring does not survive the extra dimension. The body-centred hypercubic lattice is not a centred hypercube at all: its body centre is exactly as far away as an edge, and it is D₄, with three times the symmetry.

lattices · Centring Every six-dimensional lattice with a seven-fold rotation, on one torus. The lattices with a rotation of order seven in six dimensions are the ring ℤ[ζ₇] with three positive weights, one on each complex line it spans. Up to scale and isometry the weights live on a torus, drawn as a sixty-degree rhombus with opposite sides glued, shaded by how well each lattice packs spheres. The Galois automorphism turns the torus a third of a turn about three points, on the long diagonal: A₆* with 10080 symmetries, A₆ with 10080, and the ideal (1 − ζ)² with 672, the densest lattice in the family. The small dots are integral members, each with 28 symmetries.

A seven-fold lattice lives on a folded torus

In six dimensions a lattice can have a seven-fold rotation, and every one that does is the integers of the seventh roots of unity with three weights on it. Up to scale the weights live on a hexagonal torus that a Galois symmetry folds by a third-turn, and the fold has three fixed points: A₆*, A₆, and a third lattice that packs better than either. Centring by sevenths is no longer a reflection, as fifths was in four dimensions. It moves every lattice a third of the way along the torus and carries the three special ones round in a cycle.

lattices · Centring The five special eleven-fold lattices in ten dimensions. The five lattices at the fixed points of the Galois rotation of order five on the 4-torus of eleven-fold lattices: A₁₀*, A₁₀ and the ideals (1 − ζ)², (1 − ζ)³ and (1 − ζ)⁴ with the trace form. For each, where it sits on the torus's diagonal, its symmetry group (the root lattices' by identification, the others' by exhaustive search: 2,640 each), its kissing number, its centre density at determinant one (0.0020, 0.0094, 0.0274, 0.0189, 0.0221), and which of the five its dual is. The densest, (1 − ζ)², is shaded; the one duality fixes is (1 − ζ)³.

The densest eleven-fold lattice is not its own dual

Every ten-dimensional lattice with an eleven-fold rotation is one ring with five weights, and its shapes form a four-dimensional torus folded by a Galois rotation of order five. The seven-fold essay predicted five special lattices, the powers of 1 − ζ, cycled by centring, and all of that holds. What does not carry over is its best result. In six dimensions the densest seven-fold lattice was the one duality fixes. In ten, duality fixes (1 − ζ)³, and the densest is (1 − ζ)², whose dual is (1 − ζ)⁴.

lattices · Centring The six special thirteen-fold lattices in twelve dimensions. The six lattices at the fixed points of the Galois rotation of order six on the 5-torus of thirteen-fold lattices: A₁₂*, A₁₂ and the ideals (1 − ζ)² to (1 − ζ)⁵ with the trace form. For each, where it sits on the line through the fixed points, the symmetries known for it, its kissing number (26, 156, 390, 312, 78, 26), its centre density at determinant one (0.0005, 0.0043, 0.0213, 0.0187, 0.0081, 0.0071), and which of the six its dual is. None is its own dual. The densest, (1 − ζ)², is shaded; the densest known twelve-dimensional lattice, Coxeter and Todd's, is given below for comparison.

Thirteen-fold, and nothing is its own dual

Every twelve-dimensional lattice with a thirteen-fold rotation is one ring with six weights, and its shapes form a five-dimensional torus folded six ways. The fold has six fixed points, the powers of 1 − ζ, and duality fixes none of them, as in four dimensions and unlike six and ten: it pairs them off. The eleven-fold essay asked whether a longer cycle would let a higher power take over as the densest. It does not. (1 − ζ)² is the densest member at seven, at eleven and at thirteen, whatever duality does.

lattices · Centring The 8 special 17-fold lattices in 16 dimensions. The 8 lattices at the fixed points of the Galois fold on the torus of 17-fold lattices: A16*, A16 and the ideals (1 − ζ)² to (1 − ζ)⁷ with the trace form. For each, its place on the torus, its minimum where it reaches the bound 2kp, its kissing number (34, 272, 952, 1088, 476, 816, 306, 34), its centre density at determinant one (0.0000, 0.0009, 0.0143, 0.0215, 0.0126, 0.0191, 0.0112, 0.0007), and which of them its dual is. The densest, (1 − ζ)³, is shaded: the third power, where the seven-, eleven- and thirteen-fold families had the second.

At seventeen the densest power of 1 − ζ moves

Three primes running, the densest lattice with a p-fold rotation sat at the second power of 1 − ζ. At seventeen, in sixteen dimensions, and at nineteen, in eighteen, it sits at the third. The move is not a surprise once the minima are read: every power up to the fourth reaches Craig's bound exactly, which turns each density into a formula and the competition between powers into an inequality. The second power wins exactly while (3/2) raised to half of p − 1 stays below p, and the formula's next prediction, the fourth power at twenty-three, comes true by three per cent.

lattices · Centring A shortest vector of a sixteen-dimensional lattice, as a polynomial with two full fibres. The values of f(t) = t(t − 1)(t − 2)(t − 3) at every t in the field of 17 elements, value against t. It takes the value 0 at the four points 0, 1, 2, 3 and the value 12 at the four points 3, 5, 12, 14. Those two fibres are two sets of four with equal first, second and third power sums modulo 17, so the vector with +1 on one set and −1 on the other lies in (1 − ζ)⁴, with squared length 8 in the trace form's units of 17: Craig's bound, attained.

The shortest vector is a polynomial with two full fibres

Whether Craig's lattices reach their bound, minimum 2kp for the ideal (1 − ζ)ᵏ, looked like a question for shortest-vector searches in twenty-eight to thirty-six dimensions. It is a question about polynomials over the field of p elements. The bound is reached exactly when some polynomial of degree k takes two values k times each, and every divisor of p − 1 supplies one. Read that way, every power at 29, 31 and 37 is settled, the densest are the fourth, fourth and fifth as predicted, and past the middle of each cycle the minimum climbs until it meets the vector of quadratic residues and stops.

lattices · Centring

Named alongside it

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

CentringKissing numberCyclotomic polynomialsDual latticeBravais latticeCrystallographic restrictionHolohedryQuasicrystalArithmetic crystal classClassificationCompanion matrixCut-and-project

All concepts