The Symmetry of Densest Packings of Space

University Of Texas At Austin, Austin TX

Investigators

Abstract

This proposal is in the spirit of Hilbert's Eighteenth problem, to study the symmetries of the optimally dense packings, by spheres or polyhedra, of Euclidean and hyperbolic spaces of general dimension. Very few particular such packing problems have ever been solved, but our goal is more directed to study such problems as a class, to specify them so that they are well-posed, with reasonable conditions for existence and uniqueness, and in particular to then study the symmetries of the solutions. This work is motivated in part by the discovery of aperiodic tilings, such as the Penrose tilings of the plane by the kite and dart polygons, tilings which can be understood as solutions of such a packing problem. The symmetries of aperiodic tilings have long been connected with the mathematics of ergodic theory, using either the translation group or the full congruence group of the space being tiled as the dynamics. The symmetry of the tilings has then been related to the conjugacy class of the associated dynamical system. Several questions are proposed here about the symmetry of packing problems, some directed at general qualitative behavior and some directed at important special cases. For instance, it is proposed to show that the symmetry of the densest packings of a hyperbolic space by spheres of fixed radius is different for different radii. (It is already known that for most radii the densest packings are aperiodic - they cannot have crystallographic symmetry.) And more generally it is proposed to show that a "generic" packing problem, in Euclidean or hyperbolic space, only has optimal solutions which are not crystallographic. Anyone who has tried to squeeze as many pennies as possible onto a tabletop has seen that the most efficient arangement is also very symmetrical, with six pennies surrounding each. The similar problem for efficient packings of spheres in space also leads to high symmetry. But there is almost nothing known about precisely why, in general, efficiency leads to symmetry, and what kinds of symmetry are possible. Twenty years ago a new metallic alloy was discovered, a physical solution to a closely related optimization problem, and the alloy was found to possess a symmetry the nature of which is much less obvious, or, put another way, in which the mathematics of the symmetry is less well developed. This proposal concerns the study of the symmetries of efficient arrangements in space of spheres and polyhedra and in particular the development of a mathematical formalism in which such symmetries can be usefully analyzed. Particular questions about the nature of efficient packings of spheres are also specified.