# Petrial cube

Petrial cube
Rank3
TypeRegular
Notation
Schläfli symbol${\displaystyle \{4,3\}^{\pi }}$
${\displaystyle \{6,3\}_{4}}$
${\displaystyle \left\{{\frac {6}{1,3}},3:4\right\}}$
Elements
Faces4 skew hexagons
Edges12
Vertices8
Vertex figureTriangle, edge length ${\displaystyle {\sqrt {2}}}$
Petrie polygons6 squares
Related polytopes
ArmyCube
RegimentCube
Dualdouble cover of tetrahedron (degenerate)
Petrie dualCube
κ ?Tetrahedron
Convex hullCube
Abstract & topological properties
Flag count48
Euler characteristic0
Schläfli type{6,3}
SurfaceTorus
OrientableYes
Genus1
Properties
SymmetryB3, order 48
ConvexNo
Net count6
Dimension vector(1,2,2)

The Petrial cube is a regular skew polyhedron. It consists of 4 skew hexagons, and it is the Petrie dual of the cube. It has a Euler characteristic of 0.

## Vertex coordinates

The coordinates for the vertices of a Petrial cube with unit side length are the same of that as a cube, being located at ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).}$

## Related polyhedra

The rectification of the Petrial cube is the octahemioctahedron, which is uniform.

The Petrial cube is abstractly equivalent to the blended tetrahedron.