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

For $$d\in {\mathbb {N}}$$ 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