### 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 |
---|---|

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 |
---|---|

Volume | 27 |

ISSN (Print) | 1217-4696 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics

### 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

**Configuration spaces of equal spheres touching a given sphere : The twelve spheres problem.** / Kusner, Rob; Kusner, Wöden; Lagarias, Jeffrey C.; Shlosman, Senya.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Research › peer-review

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

}

TY - CHAP

T1 - Configuration spaces of equal spheres touching a given sphere

T2 - The twelve spheres problem

AU - Kusner, Rob

AU - Kusner, Wöden

AU - Lagarias, Jeffrey C.

AU - Shlosman, Senya

PY - 2018/1/1

Y1 - 2018/1/1

N2 - The problem of twelve spheres is to understand, as a function of r ϵ (0,rmax(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 rmax(N) is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.

AB - The problem of twelve spheres is to understand, as a function of r ϵ (0,rmax(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 rmax(N) is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.

KW - Configuration spaces

KW - Constrained optimization

KW - Criticality

KW - Discrete geometry

KW - Materials science

KW - Morse theory

UR - http://www.scopus.com/inward/record.url?scp=85056318530&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-57413-3_10

DO - 10.1007/978-3-662-57413-3_10

M3 - Chapter

T3 - Bolyai Society Mathematical Studies

SP - 219

EP - 277

BT - Bolyai Society Mathematical Studies

PB - Springer Berlin - Heidelberg

ER -