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New Trends in Intuitive GeometryContact Numbers for Sphere Packings

New Trends in Intuitive Geometry: Contact Numbers for Sphere Packings [In discrete geometry, the contact number of a given finite number of non-overlapping spheres was introduced as a generalization of Newton’s kissing number. This notion has not only led to interesting mathematics, but has also found applications in the science of self-assembling materials, such as colloidal matter. With geometers, chemists, physicists and materials scientists researching the topic, there is a need to inform on the state of the art of the contact number problem. In this paper, we investigate the problem in general and emphasize important special cases including contact numbers of minimally rigid and totally separable sphere packings. We also discuss the complexity of recognizing contact graphs in a fixed dimension. Moreover, we list some conjectures and open problems.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

New Trends in Intuitive GeometryContact Numbers for Sphere Packings

Part of the Bolyai Society Mathematical Studies Book Series (volume 27)
Editors: Ambrus, Gergely; Bárány, Imre; Böröczky, Károly J.; Fejes Tóth, Gábor ; Pach, János
Bezdek, Károly; Khan, Muhammad A.
New Trends in Intuitive Geometry — Nov 3, 2018
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22 pages

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Publisher
Springer Berlin Heidelberg
Copyright
© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018. Corrected Publication 2018
ISBN
978-3-662-57412-6
Pages
25 –47
DOI
10.1007/978-3-662-57413-3_2
Publisher site
See Chapter on Publisher Site

Abstract

[In discrete geometry, the contact number of a given finite number of non-overlapping spheres was introduced as a generalization of Newton’s kissing number. This notion has not only led to interesting mathematics, but has also found applications in the science of self-assembling materials, such as colloidal matter. With geometers, chemists, physicists and materials scientists researching the topic, there is a need to inform on the state of the art of the contact number problem. In this paper, we investigate the problem in general and emphasize important special cases including contact numbers of minimally rigid and totally separable sphere packings. We also discuss the complexity of recognizing contact graphs in a fixed dimension. Moreover, we list some conjectures and open problems.]

Published: Nov 3, 2018

Keywords: Sphere packings; Kissing number; Contact numbers; Totally separable sphere packings; Minimal rigidity; Rigidity; Erdős-type distance problems; Colloidal matter; (Primary) 52C17; 52C15; (Secondary) 52C10

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