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

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