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The lattice covering density of the regular octahedron
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 32/27k3+O(k2) as k→∞.
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BibTeX
@misc{lian2026lattice,
title = {{The lattice covering density of the regular octahedron}},
author = {Yanlu Lian and Fei Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23210},
eprint = {2609.23210},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23210},
}
RIS
TY - UNPB
TI - The lattice covering density of the regular octahedron
AU - Yanlu Lian
AU - Fei Xue
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23210
UR - https://arxiv.org/abs/2609.23210
AB - 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 32/27k^3+O(k^2) as k→∞.
ER -