Crystallography & Symmetry
patterns derived from their groups
Series

Lengths — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · lattices
  2. 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.

    part 2 · lattices
  3. Both sides of the transformation, on five lattices. A Gaussian of width set by t on every point of a lattice, summed; and the same sum over the dual lattice with the width inverted and the covolume divided out. The two agree to the last bit a double carries, at every t and on lattices with no symmetry in them, so nothing here is a coincidence of parameters. The identity is exact and the reason to have it is that the two sides do not cost the same.

    The sum that turns a lattice into its dual

    Put a Gaussian on every point of a lattice and add them up. The answer equals the same sum over the dual lattice with the width inverted and the covolume divided out — exactly, to the last bit a double carries, on lattices with no symmetry in them. The identity is free and the reason to have it is that the two sides do not cost the same: at one end of the range the direct sum needs forty thousand terms and the dual sum needs a hundred and twenty-five.

    part 3 · lattices
  4. The same terms, added in two shapes. Partial sums of the alternating 1/r sum over the simple cubic lattice, taken over expanding cubes and over expanding spheres. The terms are identical and only the order differs. The cubes creep towards 1.747565 — 1.7258 by the last point drawn — and the spheres do not settle at all, landing at -3.527 after passing through values on both sides of it. A conditionally convergent sum has no value until the order is named.

    The sum whose answer depends on the shape

    Give the points of a cubic lattice alternating signs and add up one over the distance. Added over expanding cubes the total creeps towards 1.747565; added over expanding spheres it does not converge at all, landing on both sides of that number and never settling. The terms are identical and only the order differs. Splitting the sum in two with the theta transformation gives it a value — ten decimal places from a few thousand terms.

    part 4 · lattices
  5. Face- and body-centred cubic change places at t = 1. The Gaussian sum of the face-centred cubic lattice minus that of the body-centred cubic lattice, both at unit volume, across widths from 0.35 to 2.8. Below t = 1 the difference is positive, so the body-centred lattice has the smaller sum; above, it is negative and the face-centred lattice does. At t = 1 the two are equal to the last digit a computer carries, and not by accident: at unit volume each lattice's sum at width t is the other's at width 1/t times a known factor, and at t = 1 the factor is one. Neither lattice is best at every width, and no other lattice can be either.

    No lattice in space wins at every width

    Spread a Gaussian over every point of a lattice and add them up. In the plane one lattice gives the smallest sum at every width — the hexagonal one — and that is a theorem. In space no lattice does, and the proof is two lines: at large widths the winner must be the densest lattice, at small widths it must be the densest lattice's dual, and in space those are face- and body-centred cubic. They change places at exactly the width where each is the other's dual.

    part 5 · lattices
  6. 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.

    part 6 · lattices
  7. 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.

    part 7 · lattices
  8. The window in which A₄ and its dual are both minima. The exact second derivative of the Gaussian sum at A₄ and at A₄ along their five-dimensional family of changes of shape, for widths within six hundred-thousandths of one. A₄'s turns positive at t = 0.99998110 and A₄'s turns negative at 1.00001890, the reciprocal. Between them both are positive, so both lattices are local minima; at t = 1 itself both equal 8.59e-6.

    The self-dual width is not flat

    A search over four-dimensional lattices found that A₄ and its dual stop being traps for the Gaussian sum almost exactly at the self-dual width, and read that as a flat of the sum sitting on the width itself. Computed exactly rather than by finite differences, the curvature there is not zero. It is six parts in a million of the other curvature, positive, and the same for both lattices, because duality makes their Hessians one matrix at that width. Both are minima, in a window four parts in a hundred thousand wide, with basins a few millionths across.

    part 8 · lattices

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