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New Trends in Intuitive GeometryConfiguration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem

New Trends in Intuitive Geometry: Configuration Spaces of Equal Spheres Touching a Given Sphere:... [The problem of twelve spheres is to understand, as a function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \in (0,r_{max}(12)]$$\end{document}, the configuration space of 12 non-overlapping equal spheres of radius r touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also presents general results on configuration spaces of N spheres of radius r touching a central unit sphere, with emphasis on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 \le N \le 14$$\end{document}. The problem of determining the maximal radius \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{max}(N)$$\end{document} is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

New Trends in Intuitive GeometryConfiguration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem

Part of the Bolyai Society Mathematical Studies Book Series (volume 27)
Editors: Ambrus, Gergely; Bárány, Imre; Böröczky, Károly J.; Fejes Tóth, Gábor ; Pach, János
Kusner, Rob; Kusner, Wöden; Lagarias, Jeffrey C.; Shlosman, Senya
New Trends in Intuitive Geometry — Nov 3, 2018
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58 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
219 –277
DOI
10.1007/978-3-662-57413-3_10
Publisher site
See Chapter on Publisher Site

Abstract

[The problem of twelve spheres is to understand, as a function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \in (0,r_{max}(12)]$$\end{document}, the configuration space of 12 non-overlapping equal spheres of radius r touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also presents general results on configuration spaces of N spheres of radius r touching a central unit sphere, with emphasis on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 \le N \le 14$$\end{document}. The problem of determining the maximal radius \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{max}(N)$$\end{document} is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.]

Published: Nov 3, 2018

Keywords: Configuration spaces; Discrete geometry; Morse theory; Constrained optimization; Criticality; Materials science; 11H31; 49K35; 52C17; 52C25; 53C22; 55R80; 57R70; 58E05; 58K05; 70G10; 82B05

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