A quaternionic construction behind $841$-point kissing arrangement in ℝ¹²

@misc{takhanov2026quaternionic,
  title = {{A quaternionic construction behind $841$-point kissing arrangement in $\mathbb R^12$}},
  author = {Rustem Takhanov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09179},
  eprint = {2609.09179},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09179},
}

TY  - UNPB
TI  - A quaternionic construction behind $841$-point kissing arrangement in ℝ^12
AU  - Rustem Takhanov
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.09179
UR  - https://arxiv.org/abs/2609.09179
AB  - Recently, a new record kissing arrangement of $841$ points in 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 ℝ^12≅ 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 ℍ×{0}×{0}, {0}×ℍ×{0}, {0}×{0}×ℍ, together with two $384$-point families obtained by lifting affine sets of the form {(u,v,w)∈ (𝔽_2^2)^3| u+v+w=η}, 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 ℝ^4k, k≤ 5, admit a quaternionic construction.
ER  -