# Petrial tetrahedron

Petrial tetrahedron
Rank3
TypeRegular
Notation
Schläfli symbol${\displaystyle \{3,3\}^\pi}$
${\displaystyle \{4,3\}_3}$
${\displaystyle \{4,3\mid*2\}}$
Elements
Faces3 skew squares
Edges6
Vertices4
Vertex figureTriangle, edge length 1
Petrie polygons4 triangles
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt6}{4} \approx 0.61237}$
Related polytopes
ArmyTet
RegimentTet
DualHemioctahedron (abstract)
Petrie dualTetrahedron
Orientation double coverCube
Abstract properties
Flag count24
Net count1
Euler characteristic1
Schläfli type{4,3}
Topological properties
SurfaceReal projective plane
OrientableNo
Genus1
Properties
SymmetryA3, order 24

The Petrial tetrahedron is a regular skew polyhedron. It is composed of 3 skew squares It is the Petrie dual of the regular tetrahedron.

## Hemicube

Hemicube

The hemicube is a tiling of the real projective plane which is abstractly equivalent to the Petrial tetrahedron. It's double cover is a cube and it can be seen as a cube with antipodal points identified. In other words it a quotient of the cube.

The hemicube is also related to the triangular hosohedron, as if the opposite edges and vertices of each square are identified, the result is a triangular hosohedron.

## Related polyhedra

The rectified petrial tetrahedron is the tetrahemihexahedron, a uniform polyhedron.