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Computing Upper Bounds for the Packing Density of Congruent Copies of a Convex Body

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New Trends in Intuitive Geometry

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 27))

Abstract

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in \(\mathbb {R}^n\); this theorem is a generalization of the linear programming bound for sphere packings. We illustrate its use by computing an upper bound for the maximum density of packings of regular pentagons in the plane. Our computational approach is numerical and uses a combination of semidefinite programming, sums of squares, and the harmonic analysis of the Euclidean motion group. We show how, with some extra work, the bounds so obtained can be made rigorous.

The first author was support by Rubicon grant 680-50-1014 from the Netherlands Organization for Scientific Research (NWO). The second author was supported by Vidi grant 639.032.917 from the Netherlands Organization for Scientific Research (NWO).

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Notes

  1. 1.

    On the five equilateral bodies, that are usually called regular, and which of them fill their natural space, and which do not, in contradiction to the commentator of Aristotle, Averroës.

  2. 2.

    Below I show these angles with their chords:

    Angle of the pyramid – 70 degrees. 31 minutes. \(43\frac{1}{2}\) seconds. chord 1154701.

  3. 3.

    A lattice is a discrete subgroup of \((\mathbb {R}^n, {+})\).

  4. 4.

    Like all the graphs considered in this paper.

  5. 5.

    http://maurolico.free.fr.

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Acknowledgements

We are thankful to Pier Daniele Napolitani and Claudia Addabbo from the Maurolico Project, who provided us with a transcript of Maurolico’s manuscript. In particular, Claudia Addabbo provided us with a draft of her commented Italian translation of the manuscript.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, TU Delft, Van Mourik Broekmanweg 6, 2628, XE Delft, The Netherlands

    Fernando Mário de Oliveira Filho

  2. Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931, Köln, Germany

    Frank Vallentin

Authors
  1. Fernando Mário de Oliveira Filho
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  2. Frank Vallentin
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Corresponding author

Correspondence to Fernando Mário de Oliveira Filho .

Editor information

Editors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gergely Ambrus

  2. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Károly J. Böröczky

  4. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gábor Fejes Tóth

  5. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    János Pach

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de Oliveira Filho, F.M., Vallentin, F. (2018). Computing Upper Bounds for the Packing Density of Congruent Copies of a Convex Body. In: Ambrus, G., Bárány, I., Böröczky, K., Fejes Tóth, G., Pach, J. (eds) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_7

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