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

For d∈ N d ∈ N , a compact sphere packing of Euclidean space Rᵈ R d is a set of spheres in Rᵈ 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 Rᵈ 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<r₀<r₁<⋯ <rₙ₋₁=1 0 < r 0 < r 1 < ⋯ < r n - 1 = 1 can occur as the radii of spheres in a compact sphere packing of Rᵈ 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 Rᵈ R d which have exactly n sizes of sphere and which are associated to the fixed heteroperturbative set.