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

The problem of twelve spheres is to understand, as a function of $r \in (0,r_{max}(12)]$,theconfigurationspaceof$12$ 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 L'aszl'o Fejes T'oth made significant contributions.