On the Sausage Catastrophe in 4 Dimensions

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 $n_d^{\ast }$ be the smallest value of $n$ for which the densest packing of $n$ spheres in ${\mathbb{R}}^d$ is full-dimensional and $N_d^{\ast }$ be the smallest value of $N$ for which the densest packing of $N$ spheres in ${\mathbb{R}}^d$ is full-dimensional for all $N\ge N_d^{\ast }$. We extend the work of Gandini and Zucco (1992) to obtain new upper bounds of $n_4^{\ast }\le 338,196$ and $N_4^{\ast }\le 516,946$. Some lengthy and repetitive components of the proof of the latter result were obtained using interval arithmetic.

31 pages (21 pages + 10 page appendix), 13 figures