# Hemiicosahedron

Hemiicosahedron Rank3
TypeRegular
Space5-dimensional Euclidean space
Notation
Bowers style acronymEllike
Schläfli symbol$\{3,5\}_5$ Elements
Faces10 triangles
Edges15
Vertices6
Vertex figurePentagonal-pentagrammic coil
Petrie polygons6 pentagonal-pentagrammic coils
Related polytopes
ArmyHix
DualHemidodecahedron (5-dimensional)
Petrie dualPetrial hemiicosahedron
Convex hullHexateron
Orientation double coverIcosahedron
Abstract & topological properties
Flag count60
Euler characteristic1
Schläfli type{3,5}
SurfaceReal projective plane
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 A fundamental domain of the hemicuboctahedron with dotted blue lines indicating where new edges can be added to form the hemiicosahedron.

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.