A Computer Approach to Determine the Densest Translative Tetrahedron Packings

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 article 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.

Keywords:

• tetrahedron packing
• octahedron packing
• color graph
• optimization

Acknowledgments

This work is supported by the 973 Program 2013CB834201 and the Chang Jiang Scholars Program of China. For helpful discussions, I am grateful to professor R. J. Gardner and professor F. Santos.