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arXiv · 1611.10297

Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem

Rob Kusner·Wöden Kusner·Jeffrey C. Lagarias·Senya Shlosman
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Abstract

The problem of twelve spheres is to understand, as a function of $r \in (0,r_{max}(12)]$, 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 $3 \le N \le 14$. The problem of determining the maximal radius $r_{max}(N)$ is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.

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Rob Kusner, Wöden Kusner, Jeffrey C. Lagarias, Senya Shlosman. 2018-03-05. Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem. https://doi.org/10.1007/978-3-662-57143-3

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