# Petrial tetrahedron

Jump to navigation Jump to search
Petrial tetrahedron
Rank3
TypeRegular
Notation
Bowers style acronymElcube
Schläfli symbol${\displaystyle \{3,3\}^{\pi }}$
${\displaystyle \{4,3\}_{3}}$
${\displaystyle \{4,3\mid *2\}}$
${\displaystyle \left\{{\frac {4}{1,2}},3:3\right\}}$
Elements
Faces3 skew squares
Edges6
Vertices4
Vertex figureTriangle, edge length 1
Petrie polygons4 triangles
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{4}}\approx 0.61237}$
Related polytopes
ArmyTet
RegimentTet
DualHemioctahedron (abstract)
Petrie dualTetrahedron
κ ?Cube
Orientation double coverCube
Abstract & topological properties
Flag count24
Euler characteristic1
Schläfli type{4,3}
SurfaceReal projective plane
OrientableNo
Genus1
Properties
SymmetryA3, order 24
Flag orbits1
ConvexNo
Net count1
Dimension vector(1,2,2)

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

## Hemicube

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