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

Abstract

AbstractFor $$d\in {\mathbb {N}}$$ d ∈ N , a compact sphere packing of Euclidean space $${\mathbb {R}}^{d}$$ R d is a set of spheres in $${\mathbb {R}}^{d}$$ 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 $${\mathbb {R}}^{d}$$ R d . We are motivated by the question: For $$d,n\in {\mathbb {N}}$$ d , n ∈ N with $$d,n\ge 2$$ d , n ≥ 2 , how many configurations of numbers $$0<r_{0}<r_{1}<\cdots <r_{n-1}=1$$ 0 < r 0 < r 1 < ⋯ < r n - 1 = 1 can occur as the radii of spheres in a compact sphere packing of $${\mathbb {R}}^{d}$$ 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 $$d,n\in {\mathbb {N}}$$ d , n ∈ N with $$d,n\ge 2$$ 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 $${\mathbb {R}}^{d}$$ R d which have exactly n sizes of sphere and which are associated to the fixed heteroperturbative set.