Hemiicosahedron

The hemiicosahedron is a regular map and abstract polytope. It is a tiling of the projective plane. It can be seen as an icosahedron with antipodal faces identified.

Related polytopes
The hemiicosahedron can be made by subdividing the square faces of the hemicuboctahedron into triangles in a particular fashion. And likewise the hemicuboctahedron can be made by combining certain triangular faces of the hemiicosahedron into squares. This yields a embedding of the hemiicosahedron in 3-dimensional Euclidean space as a subdivision of the tetrahemihexahedron. The resulting concrete polytope is isogonal but not regular under isometry.

The hemiicosahedron appears as a facet of the regular abstract polychoron, the 11-cell.

The skeleton of the hemiicosahedron is the complete graph $$K_6$$, the same skeleton as the 5-simplex. This leads to a skew embedding of the hemiicosahedron in 5-dimensional Euclidean space with the same vertices and edges as the 5-simplex. This embedding is regular under isometry.