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

TL;DR

This paper explores the configuration space of twelve equal spheres touching a central sphere, reviewing historical work, presenting new results, and discussing topology and geometry aspects relevant to physics and materials science.

Contribution

It provides new insights and results on the configuration spaces of equal spheres touching a central sphere, extending to N spheres and discussing the maximal radius problem.

Findings

New results on configuration spaces for N=12 to 14 spheres.

Formulation of conjectures related to sphere arrangements.

Analysis of the maximal radius in the context of the Tammes problem.

Abstract

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 L\'aszl\'o Fejes T\'oth made significant contributions.