Crystallography & Symmetry
patterns derived from their groups
Concept

Perfect form — where it appears

A positive quadratic form determined completely by its shortest vectors and their common length. Voronoi showed the lattices of locally densest packing are the perfect and eutactic ones; four dimensions has two, D₄ and A₄.

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

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.

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.

lattices · Packing 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 matrixLatticePoisson summationTheta seriesBravais latticeChange of basisDeterminantIrreducible representationKissing numberLattice automorphismPacking density

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