# Hemiicosahedron

Hemiicosahedron
Rank3
Dimension5
TypeRegular
Notation
Bowers style acronymEllike
Schläfli symbol${\displaystyle \{3,5\}_{5}}$
Elements
Faces10 triangles
Edges15
Vertices6
Vertex figurePentagonal-pentagrammic coil
Petrie polygons6 pentagonal-pentagrammic coils
Related polytopes
ArmyHix
DualHemidodecahedron (5-dimensional)
Petrie dualPetrial hemiicosahedron
φ 2 Hemi great dodecahedron
Convex hullHexateron
Orientation double coverIcosahedron
Abstract & topological properties
Flag count60
Euler characteristic1
Schläfli type{3,5}
SurfaceReal projective plane[1]
OrientableNo
Genus1
SkeletonK6

The hemiicosahedron is a regular abstract polytope and regular skew polyhedron in 5-dimensional Euclidean space. As an abstract polytope, it is a tiling of the projective plane and can be seen as an icosahedron with antipodal faces identified. It has a single faithful realization in 5-dimensional Euclidean space.

## Vertex coordinates

The vertex coordinates of the hemiicosahedron in 5-dimensional Euclidean space are the same as the those of the hexateron.

## 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 ${\displaystyle K_{6}}$, 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.