Upper bound on the packing density of regular tetrahedra and octahedra

Details Upper bound on the packing density of regular tetrahedra and octahedra Upper bound on the packing density of regular tetrahedra and octahedra

2010-08-17

arxiv.org/abs/1008.2830v1

Discrete and computational geometry, 2010

Mathematics

Metric Geometry

21 pages, 4 figures

Scientific paper

10.1007/s00454-010-9304-x

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.

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