The twelve spheres problem

The problem of $12$ 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 moved around on the unit sphere, 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 addresses results on configuration spaces of $N$ spheres of radius $r$ touching a central unit sphere, for $3 \le N\le 14$. The problem of determining the maximal radius $r_{max}(N)$ is equivalent to the Tammes problem, to which L\'{a}szl\'{o} Fejes T\'{o}th made significant contributions.