2020/03/20 by Arthur Baragar, Baragar, Arthur
Computer Science · Mathematics · #11H31 #14J28 #20F55 #20H15 #52C17 #52C26 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2003.09112
openalex publication_date 2020/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We call a packing of hyperspheres in n dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the n-dimensional space; and every sphere in the packing is a member of a cluster of n+2 mutually tangent spheres (and a few more properties described herein). In this paper, we describe an Apollonian packing in eight dimensions that naturally arises from the study of generic nodal Enriques surfaces. The E7, E8 and Reye lattices play roles. We use the packing to generate an Apollonian packing in nine dimensions, and a cross section in seven dimensions that is weakly Apollonian. Maxwell described all three packings but seemed unaware that they are Apollonian. The packings in seven and eight dimensions are different than those found in an earlier paper. In passing, we give a sufficient condition for a Coxeter graph to generate mutually tangent spheres, and use this to identify an Apollonian sphere packing in three dimensions that is not the Soddy sphere packing.