Related families of polytopes The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions. The hypercube (offset) family is one of three regular polytope families, labeled by Coxeter as γn. The other two are the hypercube dual family, the cross-polytopes, labeled as βn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn. Another related family of semiregular and uniform polytopes is the demihypercubes, which are constructed from hypercubes with alternate vertices deleted and simplex facets added in the gaps, labeled as hγn. n-cubes can be combined with their duals (the cross-polytopes) to form compound polytopes: In two dimensions, we obtain the octagrammic star figure {8/2}, In three dimensions we obtain the compound of cube and octahedron, In four dimensions we obtain the compound of tesseract and 16-cell. Relation to (n−1)-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. Generalized hypercubes Regular complex polytopes can be defined in complex Hilbert space called generalized hypercubes, γ = p{4}2{3}...2{3}2, or ... Real solutions exist with p = 2, i.e. γ = γn = 2{4}2{3}...2{3}2 = {4,3,..,3}. For p > 2, they exist in . The facets are generalized (n−1)-cube and the vertex figure are regular simplexes. The regular polygon perimeter seen in these orthogonal projections is called a petrie polygon. The generalized squares (n = 2) are shown with edges outlined as red and blue alternating color p-edges, while the higher n-cubes are drawn with black outlined p-edges. The number of m-face elements in a p-generalized n-cube are: . This is pn vertices and pn facets. Relation to exponentiation Any positive integer raised to another positive integer power will yield a third integer, with this