On Edge's correspondence associated with \cdot 222

Abstract: We describe explicitly the correspondence of Edge between the set of planes contained in the Fermat cubic 4-fold in characteristic 2, and the set of lattice points T of the Leech lattice such that OABT is a regular tetrahedron, where O is the origin of the Leech lattice, and A and B are fixed points of the Leech lattice such that OAB is a regular triangle of edge length 2. Using this description, we present Conway's isomorphism from PSU(6,4) to \cdot 222 in terms of matrices.