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arXiv · 2609.23210

The lattice covering density of the regular octahedron

Yanlu Lian·Fei Xue
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Abstract

We prove that the lattice covering density of the regular octahedron is $9/8$, establishing the optimality of the classical body-centered cubic covering. A hypothetical lattice covering of smaller density would yield a lattice basis whose six Kuhn tetrahedra each fit in a translate of the octahedron. A sharp determinant bound then gives a contradiction. As a consequence, the maximum order of an undirected Abelian Cayley graph with three generators and their inverses and diameter at most $k$ is $\frac{32}{27}k^3+O(k^2)$ as $k\to\infty$.

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Yanlu Lian, Fei Xue. 2026-11-01. The lattice covering density of the regular octahedron. https://arxiv.org/abs/2609.23210

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