On the Sausage Catastrophe in 4 Dimensions

TL;DR

This paper investigates the Sausage Catastrophe in 4-dimensional sphere packings, providing new upper bounds for the smallest configurations where densest packings become full-dimensional, extending previous work with computational methods.

Contribution

It extends prior research by establishing new upper bounds for key parameters in 4D sphere packings, using interval arithmetic for rigorous proof components.

Findings

New upper bound for n*_4: 338,196

New upper bound for N*_4: 516,946

Use of interval arithmetic for proof components

Abstract

The Sausage Catastrophe of J. Wills (1983) is the observation that in $d=3$ and $d=4$, the densest packing of $n$ spheres in $\mathbb{R}^{d}$ is a sausage for small values of $n$ and jumps to a full-dimensional packing for large $n$ without passing through any intermediate dimensions. Let $n_{d}^{*}$ be the smallest value of $n$ for which the densest packing of $n$ spheres in $\mathbb{R}^{d}$ is full-dimensional and $N_{d}^{*}$ be the smallest value of $N$ for which the densest packing of $N$ spheres in $\mathbb{R}^{d}$ is full-dimensional for all $N\geq N_{d}^{*}$. We extend the work of Gandini and Zucco (1992) to obtain new upper bounds of $n_{4}^{*}\leq338,\!196$ and $N_{4}^{*}\leq516,\!946$. Some lengthy and repetitive components of the proof of the latter result were obtained using interval arithmetic.