A note on five dimensional kissing arrangements

Abstract

The kissing number $\tau (d)$ is the maximum number of pairwise non-overlapping unit spheres each touching a central unit sphere in the $d$-dimensional Euclidean space. In this note we report on how we discovered a new, previously unknown arrangement of $40$ unit spheres in dimension $5$. Our arrangement saturates the best known lower bound on $\tau (5)$, and refutes a `belief' of Cohn--Jiao--Kumar--Torquato.

Workshop on Orthogonal designs and related Combinatorics, Meiji University, Tokyo