Hermite Constant in Dimension 9 - Mathieu Dutour Sikirić

The Lattice Packing Problem in Dimension 9

The Hermite constant γ_n characterizes the maximum density of a sphere packing in Rⁿ where the centers of the spheres form a lattice. My research, in collaboration with Wessel Van Woerden, successfully determined this constant for dimension 9.

Two dimensional hexagonal packing — 2D Case: The square and hexagonal lattice packing (A₂).

3D cannon ball packing — 3D Case: The "cannon ball" packing (HCP/FCC lattices).

Scientific Achievement

By implementing a highly optimized version of Voronoi's algorithm, we were able to classify the relevant lattices in dimension 9. This computational feat required significant algorithmic refinements to handle the exponential growth of the search space in higher dimensions.

The results confirm the value of the Hermite constant γ₉ and provide a complete classification of the extreme lattices in this dimension.

arXiv:2508.20719 M. Dutour Sikirić, W. Van Woerden, The lattice packing problem in dimension 9 by Voronoi's algorithm (2025).

Data available at Zenodo:15707640.