Hypercube (EntityClass, 17)

From Higher Dimensions Database

A hypercube is an n-dimensional polytope which is the dual of that dimension's cross polytope. They exist in all dimensions. They can be represented by the bracketopic string [a1a2...an] or by the combined Coxeter-Dynkin string x4o(3o)*.

Under the elemental naming scheme, hypercubes are denoted by the geo- prefix, meaning the classical element of "earth".

Number of hypercells in a hypercube

Dimension of hypercube Formula
0 1 2 3 4 5
(image) 0 1 2 4 8 16 32 2n
1 0 1 4 12 32 80 n2n-1
2 0 0 1 6 24 80 n(n-1)2n-22-1
3 0 0 0 1 8 40 n(n-1)(n-2)2n-36-1
4 0 0 0 0 1 10 n(n-1)(n-2)(n-3)2n-424-1
5 0 0 0 0 0 1 n(n-1)(n-2)(n-3)(n-4)2n-5120-1
Sum 1 3 9 27 81 243 3n

Number of k-cubes in an n-cube: 2n-kn!/(k!(n-k)!)


Hypercubes
pointdigonsquarecubegeochorongeoterongeopeton

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