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Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra
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  • Published: 14 October 2010

Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra

  • Simon Gravel1,
  • Veit Elser2 &
  • Yoav Kallus2 

Discrete & Computational Geometry volume 46, pages 799–818 (2011)Cite this article

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  • 12 Citations

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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.

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References

  1. Aristotle: On the Heavens, vol. III. ebooks@adelaide.edu. Translated by J.L. Stocks

  2. Bezdek, A., Kuperberg, W.: Dense packing of space with various convex solids. Preprint arXiv:1008.2398v1 (2010)

  3. Chen, E.R.: A dense packing of regular tetrahedra. Discrete Comput. Geom. 40, 214 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, E.R., Engel, M., Glotzer, S.: Dense crystalline dimer packings of regular tetrahedra. Discrete Comput. Geom. 44, 253 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conway, J.H., Torquato, S.: Packing, tiling, and covering with tetrahedra. Proc. Natl. Acad. Sci. USA 103(28), 10612 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Euclid, Heath, T.L., Heiberg, J.L.: The Thirteen Books of Euclid’s Elements: Books X–XIII and Appendix. Cambridge University Press, Cambridge (1908)

    Google Scholar 

  7. Haji-Akbari, A., Engel, M., Keys, A., Zheng, X.: Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra. Nature 462, 773 (2009)

    Article  Google Scholar 

  8. Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. Second Ser. 162, 1065 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hales, T.C., Harrison, J., McLaughlin, S., Nipkow, T., Obua, S., Zumkeller, R.: A revision of the proof of the Kepler conjecture. Discrete Comput. Geom. 44, 1 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hilbert, D.C.: Mathematische probleme. Nachr. Ges. Wiss. Gött., Math. Phys. Kl. 3, 253 (1900)

    Google Scholar 

  11. Hoylman, D.J.: The densest lattice packing of tetrahedra. Bull. Am. Math. Soc. 76, 135 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jaoshvili, A., Esakia, A., Porrati, M., Chaikin, P.M.: Experiments on the random packing of tetrahedral dice. Phys. Rev. Lett. 104, 185501 (2010)

    Article  Google Scholar 

  13. Kallus, Y., Elser, V., Gravel, S.: A method for dense packing discovery. Preprint arXiv:1003.3301 (2010)

  14. Kallus, Y., Elser, V., Gravel, S.: Dense periodic packings of tetrahedra with small repeating units. Discrete Comput. Geom. 44, 245 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Senechal, M.: Which tetrahedra fill space? Math. Mag. 54(5), 227 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Torquato, S., Jiao, Y.: Dense packings of polyhedra: Platonic and archimedean solids. Phys. Rev. E 80, 041104 (2009)

    Article  MathSciNet  Google Scholar 

  17. Torquato, S., Jiao, Y.: Dense packings of the platonic and archimedean solids. Nature 460, 876 (2009)

    Article  Google Scholar 

  18. Torquato, S., Jiao, Y.: Exact constructions of a family of dense periodic packings of tetrahedra. Phys. Rev. E 81, 041310 (2010)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Genetics, Stanford University School of Medicine, Stanford, CA, 94305-5120, USA

    Simon Gravel

  2. Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, 14853-2501, USA

    Veit Elser & Yoav Kallus

Authors
  1. Simon Gravel
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  2. Veit Elser
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  3. Yoav Kallus
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Correspondence to Simon Gravel.

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Gravel, S., Elser, V. & Kallus, Y. Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra. Discrete Comput Geom 46, 799–818 (2011). https://doi.org/10.1007/s00454-010-9304-x

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  • Received: 16 August 2010

  • Accepted: 28 September 2010

  • Published: 14 October 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00454-010-9304-x

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Keywords

  • Tetrahedron
  • Octahedron
  • Packing
  • Upper bound
  • Regular solid
  • Hilbert problem
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