Crystallography & Symmetry
patterns derived from their groups
Concept

Poisson summation — where it appears

The identity that a function summed over a lattice equals its Fourier transform summed over the dual lattice, divided by the covolume. For a Gaussian it ties a lattice's sum at one width to its dual's at the reciprocal width.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

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 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.

lattices · Lengths

Named alongside it

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

Dual latticeGram matrixTheta seriesLatticePerfect formIrreducible representationKissing numberLattice automorphismLattice sumReciprocal latticeSphere packing

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