# 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 | K_{6} |

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

This article is a stub. You can help Polytope Wiki by expanding it. |