Hypercube (EntityClass, 17)
From Hi.gher. Space
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 | |||
| 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 |
| point • digon • square • cube • geochoron • geoteron • geopeton |
