On Compact Packings of Euclidean Space with Spheres of Finitely Many Sizes

Abstract For d {N} d ∈ N, a compact sphere packing of Euclidean space {R}^d R d is a set of spheres in {R}^d R d with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous simplicial d -complex that covers all of {R}^d R d. We are motivated by the question: For d, n {N} d, n ∈ N with d, n 2 d, n ≥ 2, how many configurations of numbers 0 0 r 0 r 1 ⋯ r n - 1 = 1 can occur as the radii of spheres in a compact sphere packing of {R}^d R d wherein there occur exactly n sizes of sphere? We introduce what we call ‘heteroperturbative sets’ of labeled triangulations of unit spheres and we discuss the existence of non-trivial examples of heteroperturbative sets. For a fixed heteroperturbative set, we discuss how a compact sphere packing may be associated to the heteroperturbative set or not. We proceed to show, for d, n {N} d, n ∈ N with d, n 2 d, n ≥ 2 and for a fixed heteroperturbative set, that the collection of all configurations of n distinct positive numbers that can occur as the radii of spheres in a compact packing is finite, when taken over all compact sphere packings of {R}^d R d which have exactly n sizes of sphere and which are associated to the fixed heteroperturbative set.