### Abstract

The problem of twelve spheres is to understand, as a function of r ϵ (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 ≤ N ≤ 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.

Original language | English |
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Title of host publication | Bolyai Society Mathematical Studies |

Publisher | Springer Berlin - Heidelberg |

Pages | 219-277 |

Number of pages | 59 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

### Publication series

Name | Bolyai Society Mathematical Studies |
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Volume | 27 |

ISSN (Print) | 1217-4696 |

### Keywords

- Configuration spaces
- Constrained optimization
- Criticality
- Discrete geometry
- Materials science
- Morse theory

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Bolyai Society Mathematical Studies*(pp. 219-277). (Bolyai Society Mathematical Studies; Vol. 27). Springer Berlin - Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_10