Petrial tetrahedron
(Redirected from Hemicube)
| Petrial tetrahedron | |
|---|---|
 | |
| Rank | 3 |
| Type | Regular |
| Notation | |
| Bowers style acronym | Elcube |
| Schläfli symbol | |
| Elements | |
| Faces | 3 skew squares |
| Edges | 6 |
| Vertices | 4 |
| Vertex figure | Triangle, edge length 1 |
| Petrie polygons | 4 triangles |
| Measures (edge length 1) | |
| Circumradius | |
| Related polytopes | |
| Army | Tet |
| Regiment | Tet |
| Dual | Hemioctahedron (abstract) |
| Petrie dual | Tetrahedron |
| κ ? | Cube |
| Orientation double cover | Cube |
| Abstract & topological properties | |
| Flag count | 24 |
| Euler characteristic | 1 |
| Schläfli type | {4,3} |
| Surface | Real projective plane |
| Orientable | No |
| Genus | 1 |
| Properties | |
| Symmetry | A3, order 24 |
| Flag orbits | 1 |
| Convex | No |
| Net count | 1 |
| 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[edit | edit source]
The hemicube is a tiling of the real projective plane which is abstractly equivalent to the Petrial tetrahedron. It is the 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[edit | edit source]
The rectified petrial tetrahedron is the tetrahemihexahedron, a uniform polyhedron.
External links[edit | edit source]
- Hartley, Michael. "{4,3}*24".
- Wikipedia contributors. "Hemicube (geometry)".
- Klitzing, Richard. Abstract polytopes
- Wedd, N. The hemicube