# γ  3,3

γ 3
3

Rank3
Dimension3
TypeRegular
SpaceComplex
Notation
Coxeter diagram
Schläfli symbol3{4}2{3}2
Elements
Faces9 γ 3
2

Edges18 3-edges
Vertices27
Vertex figureTriangle
Related polytopes
Real analogTriangular trioprism
Dualβ 3
3

Abstract & topological properties
Flag count162
Properties
Symmetry3[4]2[3]2, order 162

γ 3
3

is a regular complex polyhedron. It is the simplest generalized cube other than the real cube.

## Vertex coordinates

Vertex coordinates for the generalized square γ 3
3

can be given as all permutations of:

• ${\displaystyle \left({\frac {\sqrt {3}}{3}},\,{\frac {\sqrt {3}}{3}},\,{\frac {\sqrt {3}}{3}}\right)}$,
• ${\displaystyle \left({\frac {\sqrt {3}}{3}},\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {3}}{6}}\pm {\frac {i}{2}}\right)}$,
• ${\displaystyle \left({\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {3}}{6}}\pm {\frac {i}{2}},\,-{\frac {\sqrt {3}}{6}}\pm {\frac {i}{2}}\right)}$,
• ${\displaystyle \left(-{\frac {\sqrt {3}}{6}}\pm {\frac {i}{2}},\,-{\frac {\sqrt {3}}{6}}\pm {\frac {i}{2}},\,-{\frac {\sqrt {3}}{6}}\pm {\frac {i}{2}}\right)}$.

## Coxeter diagrams

A generalized cube γ 3
3

can be represented by the following Coxeter diagrams:

• (full symmetry)
• (3[4]2[2]3 symmetry. Prism of γ 3
2

with a 3-edge.)
• (3[2]3[2]3 symmetry. 3-edge trioprim.)