Upper bound on the packing density of regular tetrahedra and octahedra

TL;DR

This paper establishes an extremely tight upper bound on how densely regular tetrahedra and octahedra can be packed, using geometric and recursive methods to analyze uncovered space within sphere sets centered on edges.

Contribution

It introduces a novel geometric approach to bound packing densities of regular tetrahedra and octahedra, applicable to other polyhedra with similar properties.

Findings

Upper bound on tetrahedron packing density: 2.6×10^{-25}

Upper bound on octahedron packing density: 1.4×10^{-12}

Method based on disjoint spheres on edges and solid angle analysis

Abstract

We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is not fully covered by the packing. The bound on the amount of space that is not covered in each sphere is obtained in a recursive way by building on the observation that non-overlapping regular tetrahedra cannot subtend a solid angle of $4\pi$ around a point if this point lies on a tetrahedron edge. The proof can be readily modified to apply to other polyhedra with the same property. The resulting lower bound on the fraction of empty space in a packing of regular tetrahedra is $2.6\ldots\times 10^{-25}$ and reaches $1.4\ldots\times 10^{-12}$ for regular octahedra.