Petrial cube

Petrial cube
Rank	3
Type	Regular
Notation
Schläfli symbol	$\{4,3\}^{\pi }$ $\{6,3\}_{4}$ $\left\{{\frac {6}{1,3}},3:4\right\}$
Elements
Faces	4 skew hexagons
Edges	12
Vertices	8
Vertex figure	Triangle, edge length ${\sqrt {2}}$
Petrie polygons	6 squares
Related polytopes
Army	Cube
Regiment	Cube
Dual	double cover of tetrahedron (degenerate)
Petrie dual	Cube
κ ?	Tetrahedron
Convex hull	Cube
Abstract & topological properties
Flag count	48
Euler characteristic	0
Schläfli type	{6,3}
Surface	Torus
Orientable	Yes
Genus	1
Properties
Symmetry	B3, order 48
Convex	No
Net count	6
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 $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).$

Related polyhedra

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

The Petrial cube is abstractly equivalent to the blended tetrahedron.

External links

• Wikipedia contributors. "Petrie dual".
• Hartley, Michael. "{6,3}*48".
• Wedd, N. {6,3}_(2,2)