Configuration spaces of equal spheres touching a given sphere: The twelve spheres problem

Rob Kusner*, Wöden Kusner, Jeffrey C. Lagarias, Senya Shlosman

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


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.

Original languageEnglish
Title of host publicationBolyai Society Mathematical Studies
PublisherSpringer Berlin - Heidelberg
Number of pages59
Publication statusPublished - 1 Jan 2018

Publication series

NameBolyai Society Mathematical Studies
ISSN (Print)1217-4696


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

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Applied Mathematics

Cite this