Experimental Study of Energy-Minimizing Point Configurations on Spheres

Abstract

Abstract. In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima.

The Contact Polytope of the Leech Lattice

“…Since it has maximum norm among all vertices, the corresponding facet is closest to the origin and has the largest possible circumsphere among all other facets of C(Λ 24 ). This solves a conjecture of Ballinger, Blekherman, Cohn, Giansiracusa, Kelly, and Schürmann [2,Sect. 3.7].…”

Self-assembled clusters of spheres related to spherical codes

“…If the function is strictly completely monotonic, then the universally optimal solution is also unique. For points on the surface of a sphere, the only known universally optimal solutions are [39,40] N = 1-4, 6, and 12; that is, a single point, antipodal points, points forming an equilateral triangle on the equator, and tetrahedral, octahedral, and icosahedral arrangement of points. For our purposes, this means certain desired point arrangements (e.g.…”

“…Restricting themselves to isotropic pair potentials and identical particles, Cohn and Kumar [42] constructed separate decreasing convex potential energy functions that have cubic (N=8) and dodecahedral (N=20) configurations as their minimum. Thus references [39,40,42] imply that to assemble certain clusters, it will be necessary to use more complicated HPs with carefully constructed potentials, including non-completely monotonic or anisotropic interactions.…”

Polyhedral representation conversion up to symmetries

“…Evidently every simplicial polytope is basis-simplicial. For perhaps less contrived examples of basis-simplicial polytopes than Example 7.2 we mention the Dirichlet-Voronoi-cells (DV-cells) of the root lattices D 3 , D 4 , and E 8 (convex hull of their shortest non-zero vectors) and some related universally optimal spherical polytopes (see [8]). Their boundaries consist of regular cross polytopes and regular simplices only and they thus have at most two orbits of basis with respect to their symmetry group (cf.…”