Mapping the Geometric Structure of the Leech Lattice

The Leech lattice occupies a central place in exceptional mathematics and coding theory, but because it is 24-dimensional, its geometric structure is difficult to visualize. We elucidate a coordinate system employed in Conway’s and Sloane’s Sphere Packings, Lattices and Groups to describe and to illustrate the environs of the Leech lattice’s deep holes but never explained in that book. By using this coordinate system for the deep holes, we are able to provide colorful diagrams for the environs of several deep holes. The most interesting of those deep holes is E₈³, and we describe a new E₈³-based coordinate system for the Leech lattice. We enumerate the Delaunay cells of the Leech lattice immediately neighboring a Delaunay cell of type E₈³, neighboring in the sense of intersecting the latter cell in a 23-dimensional face. There are 729 such faces, but fewer isometry classes of neighboring cells, including both deep holes and shallow holes. We also enumerate the Delaunay cells of the Leech lattice in close proximity to a Delaunay cell of type E₈³, close in the sense of having all vertices less than √2.2 distant from the center of E₈³; the latter’s circumradius is √2. Neither of the two classes of Delaunay cells enumerated below is a subset of the other. We next consider conjectures about the Leech lattice and the several lattices with similar properties, most especially the lattices ℤ , A₂, ℤ ², D₄, and E₈. Other lattices sharing fewer optimality properties include the Coxeter-Todd lattice K₁₂ and the Barnes-Wall lattice Λ₁₆. The properties of greatest interest are those involving various measures of high degrees of symmetry, as well as various measures of high efficiency in packing and covering. Our culminating theorem is a partial classification of lattices that have very strong properties similar to those of the Leech lattice.