10 - The Leech Lattice M24

Summary

We give a brief introduction to sphere-packing in n-dimensions and, in particular, to lattice packings. The notions of density and kissing number are explained. The 24-dimensional Leech lattice Λ is defined following Conway’s approach in his Three lectures on exceptional groups, see Conway (1971), with vectors in the lattice invariably displayed as they appear in the MOG. We explore the factor space Λ/2Λ, a 24-dimensional vector space over Z2, and show that its non-zero elements may be taken to be vectors of type 2 and 3 together with sets of 24 mutually orthogonal vectors of type 4 (and their negatives). These last elements are known as frames of reference or crosses; the six orbits on crosses under permutations of M₂₄ and sign changes on C-sets are described explicitly in the text, as are the orbits on vectors of types 2, 3 and 4. Finally, we explain how Λ can be defined in terms of a single Lorentz-type vector with 25 space-like coordinates and one time-like coordinate.