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

Abstract

The problem of twelve spheres is to understand, as a function of r ∈ (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 ≤ N ≤ 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.

Conformal capacity and polycircular domains

“…A very interesting discussion of many topics of combinatorial geometry including packing problems is given in the encyclopedic work of M. Berger [7]. The three dimensional case is much more difficult than the planar case and it is the subject of the extensive review paper [22] where topics range from optimal packing of spheres to constrained motion of small spheres on the surface of the unit sphere. For an extensive survey of potential theoretic extremal problems see [9].…”

Generalized representation stability for disks in a strip and no-k-equal spaces

“…In addition to the study of disks in [BBK13], others have studied the topology of configuration spaces of various identical rigid objects in various shapes of container, such as in [Alp17], [Dee11], and [KKLS18]. The choice of disks in a strip is geometrically simplest among the possibilities.…”

Configuration spaces of disks in an infinite strip

“…The topology of configuration spaces of particles with thickness has been studied earlier, for example in [2], [11], and [18], but not much seems to be known. Some of this past work is also inspired in part by applications to engineering, particularly motion planning for robots.…”