Hypercube (EntityClass, 17)

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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...a_n] or by the combined Coxeter-Dynkin string x4o(3o)*.

Under the Tamfang 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	Formula
ExPar: [#img] is obsolete, use [#embed] instead	0	1	2	4	8	16	32	2ⁿ
	1	0	1	4	12	32	80	n2^n-1
	2	0	0	1	6	24	80	n(n-1)2^n-22^-1
	3	0	0	0	1	8	40	n(n-1)(n-2)2^n-36^-1
	4	0	0	0	0	1	10	n(n-1)(n-2)(n-3)2^n-424^-1
	5	0	0	0	0	0	1	n(n-1)(n-2)(n-3)(n-4)2^n-5120^-1
	Sum	1	3	9	27	81	243	3ⁿ

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

Hypercubes