Crystallography & Symmetry
patterns derived from their groups
Concept

Kissing number — where it appears

How many of a lattice's vectors have the minimum length, which is how many spheres touch each one in the packing. It is a theta-series coefficient read at the first shell, so a question about touching spheres is a question about counting lattice points.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry.

How many vectors of each length

Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven.

lattices · Lengths The whole space of plane lattices, and its corner. Every plane lattice appears exactly once in this picture. Scaling changes no density, so the leading coefficient is fixed at one; reduction then confines the other two to 0 ≤ b ≤ 1 ≤ c, and every lattice has exactly one reduced form. The curves are the levels of constant density, which are parabolas — a density d needs 4c − b² to equal (π/2d)². They crowd toward the corner b = c = 1, which is the hexagonal lattice at π/√12 ≈ 0.9069; the square lattice sits on the left edge at π/4 ≈ 0.7854. The picture is a search over a region rather than over a list, which is what makes the answer a decision: there is nowhere else for a lattice to be.

The densest lattice in the plane

Which arrangement of equal discs covers the most floor is a question about infinitely many lattices, and reduction turns it into a question about a two-parameter region with a corner. The answer is at the corner, and the argument finishes.

applied · Packing How far apart points on a sphere can be kept. For each number of points, the largest smallest angle a search could find between any two of them. Two unit spheres touching a third do not overlap exactly when their contact points are 60° or more apart, so the largest count whose best arrangement still clears 60° is the kissing number. Twelve clears it with three degrees to spare and thirteen falls short by more than three. The circle column is the same problem in the plane, where the answer is exactly 360/n and needs no search at all.

The room a thirteenth sphere would need

Twelve equal spheres touch one, and whether a thirteenth could was argued in 1694 and settled in 1953. The reason it took so long is measurable: the twelve leave three and a half degrees of slack, which is enough room to look promising and not enough to use — and in the plane, where the same question has no slack at all, nobody ever argued.

applied · Packing E₈ has the smallest sum at every width. For four eight-dimensional lattices of unit covolume — D₈, its dual D₈, the cubic lattice Z⁸ and two copies of D₄ — the ratio of their Gaussian sums, less the constant term, to E₈'s, at widths from 0.2 to 5, logarithmic. Every ratio exceeds one everywhere: E₈ wins at every width against every rival. The closest approach is at t = 1, where D₈ and its dual are equal, a factor of 1.31 above E₈. D₈ is the better of the pair at wide widths and D₈ at narrow ones, and neither comes close to E₈ at either end.

In eight dimensions one lattice wins at every width

In space no lattice has the smallest Gaussian sum at every width, because the densest lattice wins when the Gaussian is narrow and its dual wins when it is wide, and fcc is not bcc. The argument fails exactly when the densest lattice is its own dual. In eight dimensions it is: E₈, with 240 shortest vectors, beats D₈, its dual, the cubic lattice and a sum of two four-dimensional lattices at every width tried, by a factor of at least 1.31, and its sum is mirror-symmetric in the logarithm of the width.

lattices · Lengths Almost every descent ends at D₄. At five widths of the Gaussian, twelve local minimisations of the lattice sum over all four-dimensional lattices of unit covolume, each from a random lattice, each marked by where it ends and labelled with the number of shortest vectors of the lattice it ends at. Fifty-nine of the sixty end at D₄, with twenty-four shortest vectors. One, at the widest Gaussian, ends at A₄*, with ten, a lattice whose sum is higher than D₄'s but which no small change lowers.

The two lattices where a four-dimensional descent stops short

In four dimensions the argument that rules out a single best lattice in space has nothing to act on, so whether one lattice wins at every width is left to be measured. A search over every four-dimensional lattice finds D₄ from almost anywhere. The exceptions are the finding: two lattices, A₄ and its dual, where a descent can stop short, each at one end of the widths and neither in the middle, for the same reason Voronoi found four dimensions has two perfect forms.

lattices · Lengths 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.

Dual latticeCentringHigher-dimensional latticeHolohedryBravais latticeSphere packingClose packingLatticeTheta seriesCoordination numberCyclotomic polynomialsPacking fraction

All concepts