发表机构
Nazarbayev University（纳扎尔巴耶夫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过四元数结构解释了$\mathbb R^{12}$中$841$点亲吻排列纪录，构造了$840$点模型并推广到$4k$维。
AI 中文摘要
最近，通过优化（Takhanov-Assylbekov-Yun，2026）在$\mathbb R^{12}$中数值上获得了$841$点的新纪录亲吻排列。该构型以坐标文件形式发布，没有其结构的数学描述。本文的目的是提供这样的描述。关键观察是，一旦我们将$\mathbb R^{12}\cong \mathbb H^3$视为四元数代数的三个副本的笛卡尔积，几何就变得透明。我们首先引入一个新的具有特定四元数结构的$840$点亲吻排列。它由三个相互正交的正则$24$胞腔组成，分别支撑在三个四元数坐标因子$\mathbb H\times\{0\}\times\{0\}$、$\{0\}\times\mathbb H\times\{0\}$、$\{0\}\times\{0\}\times\mathbb H$上，以及两个$384$点族，这些族通过将形如$$\{(u,v,w)\in (\mathbb F_2^2)^3\mid u+v+w=\eta\}$$的仿射集合提升为四元数三元组（其分量属于二元八面体群$2O$），然后应用适当的分量旋转和加权获得。该构造的一个显著特征是三个四元数因子之间存在明显的不对称性。对于移除第三个$24$胞腔后获得的$816$个向量，大部分平方范数集中在前两个四元数坐标中，而第三个坐标系统地承载较少质量。因此，第三个四维因子比前两个包含更多可用空间。然后我们证明这个$840$点构型为数值$841$点纪录提供了自然的结构模型。最后，我们引入在维数可被$4$整除时的一般四元数构造的概念，并检查${\mathbb R}^{4k}$（$k\leq 5$）中的纪录亲吻排列是否允许四元数构造。
英文摘要
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 $$\{(u,v,w)\in (\mathbb F_2^2)^3\mid u+v+w=η\},$$ 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 ${\mathbb R}^{4k}$, $k\leq 5$, admit a quaternionic construction.
