Generalized hypercube

γ p n
Rank	n
Dimension	n
Type	Regular
Space	Complex
Notation
Coxeter diagram	...
Schläfli symbol	p{4}2{3}2...2{3}2{3}2
Elements
Facets	pn  γ p n-1
Edges	2p n-1  p -edges
Vertices	p n
Vertex figure	(n -1)-simplex
Related polytopes
Dual	Generalized orthoplex
Abstract & topological properties
Flag count	$n!p^{n}$
Properties
Symmetry	p[4]2[3]2...2[3]2[3]2, order $n!p^{n}$

The generalized hypercubes, γ p
n , are a family of regular complex polytopes that generalize the hypercubes.

The generalized hypercube γ p
n  can be formed as the prism product of n  identical p -edges. This means that the real analogs of generalized hypercubes are multiprisms.

The generalized hypercubes of the form γ 2
n  have real valued coordinates and are exactly the hypercubes.

Vertex coordinates

Vertex coordinates for the generalized hypercube γ p
n  can be given as:

• $\left(e^{x_{1}2\pi i/p},e^{x_{2}2\pi i/p},e^{x_{3}2\pi i/p},\dots ,e^{x_{n}2\pi i/p}\right)$,

where each x k  is an integer ranging between 1 and p  inclusive.

External links

• Wikipedia contributors. "Hypercube#Generalized hypercubes".