An octahedron covering is a covering of three-dimensional Euclidean space by translates of an octahedron.
For a convex body 
and a full-rank point lattice 
, the density of the lattice covering
is the volume of 
divided by the volume of a fundamental
parallelepiped of 
, provided the translates 
cover Euclidean
space.
Lian and Xue (2026) proved
 |
where 
is the regular octahedron and 
denotes its lattice
covering density, settling the conjectured value. The authors report using ChatGPT
to check details of the argument, while retaining full responsibility for the mathematical
claims.
