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

  • Rob KusnerEmail author
  • Wöden Kusner
  • Jeffrey C. Lagarias
  • Senya Shlosman
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)


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.


Configuration spaces Discrete geometry Morse theory Constrained optimization Criticality Materials science 

2010 Mathematics Subject Classification

11H31 49K35 52C17 52C25 53C22 55R80 57R70 58E05 58K05 70G10 82B05 



The authors were each supported by ICERM in the Spring 2015 program on “Phase Transitions and Emergent Properties.” R. K. was also supported by the University of Pennsylvania Mathematics Department sabbatical visitor fund and by MSRI via NSF grant DMS-1440140. W. K. was also supported by Austrian Science Fund (FWF) Project 5503. J. L. was supported by NSF grant DMS-1401224 and by a Clay Senior Fellowship at ICERM. Part of the work of S. S. has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). Part of the work of S. S. has been carried out at IITP RAS. The support of Russian Foundation for Sciences (Project No. 14-50-00150) is gratefully acknowledged. The authors thank Bob Connelly, Sharon Glotzer, Mark Goresky, Tom Hales and Oleg Musin for helpful comments. Parts of Sect. 4.1 are adapted from unpublished notes by R. K. and John Sullivan (MSRI, 1994) about critical configurations of “electrons” on the sphere.


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Rob Kusner
    • 1
  • Wöden Kusner
    • 2
    • 3
  • Jeffrey C. Lagarias
    • 4
  • Senya Shlosman
    • 5
    • 6
    • 7
  1. 1.Department of Mathematics & StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Institute of Analysis and Number TheoryGraz University of TechnologyGrazAustria
  3. 3.Department of MathematicsVanderbilt UniversityNashvilleUSA
  4. 4.Department of MathematicsUniversity of MichiganAnn ArborUSA
  5. 5.Skolkovo Institute of Science and TechnologyMoscowRussia
  6. 6.Aix Marseille UniversitéUniversité de ToulonMarseilleFrance
  7. 7.Institute for Information Transmission Problems, RASMoscowRussia

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