Crystallography & Symmetry
patterns derived from their groups
Concept

Packing density — where it appears

The fraction of space filled by equal non-overlapping spheres centred on the points of a lattice, with each sphere as large as it can be. It is the sphere volume divided by the volume of one cell, and it depends only on the lattice's shape, so maximising it is a question about forms rather than about sizes.

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

How densely each shape packs, by translation alone. The densest lattice packing of each shape, as its area over the critical determinant of its difference body. The two that tile the plane by translation reach one and must, which is a check on the search rather than a result of it. The triangle reaches exactly two thirds because its difference body is a hexagon. The many-sided approximation to a circle reaches π/√12, which this collection computes a completely different way. And the pentagon is the worst of them, which is where the search is doing work nobody could do by inspection.

The densest packing of a shape that is not a disc

Which lattice packs equal discs most densely has a proof that finishes. Replace the disc with a pentagon and the same question has no closed form, but it does have a reduction: translates overlap exactly when the difference of their positions lies inside the shape minus itself, so the question becomes the smallest determinant a lattice can have while avoiding one convex body — and that is a search with a resolution attached.

applied · Packing 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

Named alongside it

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

Admissible latticeBravais latticeChange of basisConvex bodyCritical determinantDeterminantDifference bodyDual latticeGram matrixLatticeLattice packingMinkowski theorem

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