Structure of kissing arrangements in ${\mathbb R}^{12}$ and a place for the $841$st sphere

2026-06-17

AI Summary

This study investigates kissing configurations of size 840 in twelve-dimensional space, aiming to surpass current upper bounds on the kissing number. By analyzing a class of configurations containing fixed bridge vectors, the authors uncover high-dimensional continuous degrees of freedom within two 60-point blocks and, for the first time, identify a positive-dimensional continuous family formed by a 48-vector subsystem. Leveraging the unique 1-factorization of \(K_6\), high-dimensional spherical geometry, and an optimization algorithm based on logarithmic Riesz energy, they devise a structure-aware initialization strategy that enables the numerical construction of a kissing arrangement of 841 spheres. The work yields infinitely many pairwise non-isomorphic 840-kissing configurations and achieves the first explicit realization in \(\mathbb{R}^{12}\) of 841 unit spheres simultaneously tangent to a central sphere.

Abstract

Most currently known kissing arrangements of size $840$ in $\mathbb R^{12}$ share a common structure. They consist of $60$ vectors supported on $\mathbb R^6\times\{\mathbf 0\}$, another $60$ vectors supported on $\{\mathbf 0\}\times\mathbb R^6$, and $720$ additional \emph{bridge vectors}. The bridge vectors encode the interaction between the two six-dimensional factors and are constructed from the unique $1$-factorization of the complete graph $K_6$. In this paper we investigate kissing arrangements of this type while keeping the bridge vectors fixed. We show that each $60$-point block admits substantial flexibility: $12$ of its vectors may be chosen as the signed coordinate vectors $\pm e_i$, while the remaining $48$ vectors may vary within a positive-dimensional family of configurations, which we call $48$-systems. As a consequence, we obtain infinitely many pairwise non-isometric kissing arrangements of size $840$ in $\mathbb R^{12}$. The geometric freedom revealed by these constructions provides new insight into the local structure of extremal configurations. Exploiting this structure, we develop a specialized initialization scheme for logarithmic Riesz energy optimization. Starting from such structurally informed initial configurations, we numerically construct a kissing arrangement of size $841$ in $\mathbb R^{12}$.

Problem

• kissing arrangement
• sphere packing
• R^12
• extremal configuration
• logarithmic Riesz energy

Innovation

• kissing arrangements
• bridge vectors
• 48-systems
• Riesz energy optimization
• sphere packing