The dimensional family approach in (hyper)sphere packing: A typological study of new patterns, structures, and interdimensional functions

摘要

Powerful topological invariants associated with sphere packing in Euclidean spaces are investigated, and new interdimensional functions and structures are delineated that enable those invariants to be readily determined. The fundamental invariants discussed are tangency or contact number and the density of centres in the class of recursive packings, those whose process of formation is uniquely propagated throughout the entire dimensional continuum. An important property of recursivity as here defined is the maximizing of tangency, contact, or “kissing” number for a given dimension. Interdimensional functions are sought (and found) that can select maximal contact numbers even in dimensions that have many choices. Thus in 25-space there are 23 packings compatible with T24 = 196560, but only one of these is maximal, and that one our function selects. Similar properties are found with dimensions 26 (where there are at least three thousand more choices than in 25-space) through 32. In each case the function selects the maximum Tn as given in Table 1. This article is part of a book in preparation for publication.