On compact packings of Euclidean space with spheres of finitely many sizes

Eder Kikianty; Miek Messerschmidt

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arXiv · 2305.00758

On compact packings of Euclidean space with spheres of finitely many sizes

Eder Kikianty·Miek Messerschmidt

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Abstract

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 $d,n\geq2$, how many configurations of numbers $0<r_{0}<r_{1}<\ldots<r_{n-1}=1$ 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 $d,n\geq2$ 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.

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Eder Kikianty, Miek Messerschmidt. 2023-05-01. On compact packings of Euclidean space with spheres of finitely many sizes. https://arxiv.org/abs/2305.00758

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