A quaternionic construction behind $841$-point kissing arrangement in ${\mathbb{R}}^{12}$

Abstract

Recently, a new record kissing arrangement of $841$ points in ${\mathbb{R}}^{12}$ was obtained numerically by optimization (Takhanov-Assylbekov-Yun, 2026). The configuration was released as a coordinate file, without a mathematical description of its structure. The purpose of this paper is to provide such a description. The key observation is that the geometry becomes transparent once we regard ${\mathbb{R}}^{12}\cong {\mathbb{H}}^3$ as the Cartesian product of three copies of the quaternion algebra. We first introduce a new $840$-point kissing arrangement with a certain quaternionic structure. It consists of three mutually orthogonal regular $24$-cells, supported on the three quaternionic coordinate factors $\mathbb{H}\times \{0\}\times \{0\}$, $\{0\}\times \mathbb{H}\times \{0\}$, $\{0\}\times \{0\}\times \mathbb{H}$, together with two $384$-point families obtained by lifting affine sets of the form $$\{(u,v,w)\in ({\mathbb{F}}_2^2{)}^3\mid u+v+w=\eta \},$$ to quaternionic triples (whose components belong to the binary octahedral group $2O$) and then applying suitable component-wise rotations and weightings. A characteristic feature of this construction is a pronounced asymmetry among the three quaternionic factors. For the $816$ vectors obtained after removing the third $24$-cell, most of the squared norm is concentrated in the first two quaternionic coordinates, while the third coordinate carries systematically less mass. Thus, the third four-dimensional factor contains more available space than the first two. We then show that this $840$-point configuration provides a natural structural model for the numerical $841$-point record. Finally, we introduce a notion of the general quaternionic construction in dimensions divisible by $4$, and check that record kissing arrangements in ${\mathbb{R}}^{4k}$, $k\le 5$, admit a quaternionic construction.