Discrete & Computational Geometry

, Volume 46, Issue 4, pp 799–818

Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra

Authors

    • Department of GeneticsStanford University School of Medicine
  • Veit Elser
    • Laboratory of Atomic and Solid State PhysicsCornell University
  • Yoav Kallus
    • Laboratory of Atomic and Solid State PhysicsCornell University
Article

DOI: 10.1007/s00454-010-9304-x

Cite this article as:
Gravel, S., Elser, V. & Kallus, Y. Discrete Comput Geom (2011) 46: 799. doi:10.1007/s00454-010-9304-x

Abstract

Aristotle contended that (regular) tetrahedra tile space, an opinion that remained widespread until it was observed that non-overlapping tetrahedra cannot subtend a solid angle of 4π around a point if this point lies on a tetrahedron edge. From this 15th century argument, we can deduce that tetrahedra do not tile space but, more than 500 years later, we are unaware of any known non-trivial upper bound to the packing density of tetrahedra. In this article, we calculate such a bound. To this end, we show 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 solid angle argument. The argument can be readily modified to apply to other polyhedra. The resulting lower bound on the fraction of empty space in a packing of regular tetrahedra is 2.6…×10−25 and reaches 1.4…×10−12 for regular octahedra.

Keywords

TetrahedronOctahedronPackingUpper boundRegular solidHilbert problem

Copyright information

© Springer Science+Business Media, LLC 2010