  Beineke, Harary and Ringel discovered a formula for the minimum genus of a
torus in which the $n$-dimensional hypercube graph can be embedded. We give a
new proof of the formula by building this surface as a union of certain faces
in the hypercube's 2-skeleton. For odd dimension $n$, the entire 2-skeleton
decomposes into $(n-1)/2$ copies of the surface, and the intersection of any
two copies is the hypercube graph.
