# Generalized cube

Generalized cube
Rank3
Dimension3
TypeRegular
SpaceComplex
Notation
Coxeter diagram
Schläfli symbolp{4}2{3}2
Elements
Faces3p  γ p
2

Edges2p 2  p-edges
Verticesp 3
Vertex figureTriangle
Related polytopes
Real analogp -gonal trioprism
Abstract & topological properties
Flag count6p 3
Properties
Symmetryp[4]2[3]2, order 6p 3

The generalized cubes, γ p
3

, are a family of regular complex polyhedra. They generalize the cube, a real polyhedron, to complex space.

The generalized cubes are the rank-3 members of the broader class of generalized hypercubes.

## Vertex coordinates

Vertex coordinates for the generalized square γ p
3

can be given as:

• ${\displaystyle \left(e^{m2\pi i/p},e^{n2\pi i/p},e^{o2\pi i/p}\right)}$,

where m , n , and o  are integers ranging between 1 and p  inclusive.

## Coxeter diagrams

A generalized cube γ p
3

can be represented by the following Coxeter diagrams:

• (full symmetry)
• (p[4]2[2]p symmetry. Prism of a generalized square with a p -edge.)
• (p[2]p[2]p symmetry. p -edge trioprim.)