A Computer Approach to Determine the Densest Translative Tetrahedron Packings

TL;DR

This paper introduces a computational method to determine the maximum translative packing densities of solids like tetrahedra, cuboctahedra, and octahedra, addressing a longstanding geometric problem linked to Hilbert's 18th problem.

Contribution

It presents a novel computer-based approach that reduces the problem to finite optimization problems for specific solids, advancing understanding of packing densities.

Findings

Reduced packing density problems to finite optimization tasks.

Confirmed conjecture for tetrahedra, cuboctahedra, and octahedra cases.

Provided computational evidence supporting density equivalences.

Abstract

In 1900, as a part of his 18th problem, Hilbert proposed the question to determine the densest congruent (or translative) packings of a given solid, such as the unit ball or the regular tetrahedron of unit edges. Up to now, our knowledge about this problem is still very limited, excepting the ball case. It is conjectured that, for some particular solids such as tetrahedra, cuboctahedra and octahedra, their maximal translative packing densities and their maximal lattice packing densities are identical. To attack this conjecture, this paper suggests a computer approach to determine the maximal local translative packing density of a given polytope, by studying associated color graphs and applying optimization. In particular, all the tetrahedral case, the cuboctahedral case and the octahedral case of the conjecture have been reduced into finite numbers of manageable optimization problems.