Relation To n-simplices
The graph of the n-hypercube's edges is isomorphic to the Hasse diagram of the (n-1)-simplex's face lattice. This can be seen by orienting the n-hypercube so that two opposite vertices lie vertically, corresponding to the (n-1)-simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (n-1)-simplex's facets (n-2 faces), and each vertex connected to those vertices maps to one of the simplex's n-3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices.
This relation may be used to generate the face lattice of an (n-1)-simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive.
Read more about this topic: Hypercube
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