Hemiicosahedron
| Hemiicosahedron | |
|---|---|
 | |
| Rank | 3 |
| Type | Regular |
| Space | 5-dimensional Euclidean space |
| Notation | |
| Bowers style acronym | Ellike |
| Schläfli symbol | |
| Elements | |
| Faces | 10 triangles |
| Edges | 15 |
| Vertices | 6 |
| Vertex figure | Pentagonal-pentagrammic coil |
| Petrie polygons | 6 pentagonal-pentagrammic coils |
| Related polytopes | |
| Army | Hix |
| Dual | Hemidodecahedron (5-dimensional) |
| Petrie dual | Petrial hemiicosahedron |
| Convex hull | Hexateron |
| Orientation double cover | Icosahedron |
| Abstract & topological properties | |
| Flag count | 60 |
| Euler characteristic | 1 |
| Schläfli type | {3,5} |
| Surface | Real projective plane[1] |
| Orientable | No |
| Genus | 1 |
| Skeleton | K6 |
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[edit | edit source]
The vertex coordinates of the hemiicosahedron in 5-dimensional Euclidean space are the same as the those of the hexateron.
Related polytopes[edit | edit source]
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 , 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.
External links[edit | edit source]
- Hartley, Michael. "{3,5}*60".
- Klitzing, Richard. Abstract polytopes
References[edit | edit source]
Bibliography[edit | edit source]
- McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0
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