# Hemi-dodecahedron

A **hemi-dodecahedron** is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.

Hemi-dodecahedron | |
---|---|

decagonal schlegel diagram | |

Type | abstract regular polyhedron globally projective polyhedron |

Faces | 6 pentagons |

Edges | 15 |

Vertices | 10 |

Vertex configuration | 5.5.5 |

Schläfli symbol | {5,3}/2 or {5,3}_{5} |

Symmetry group | A_{5}, order 60 |

Dual polyhedron | hemi-icosahedron |

Properties | non-orientable Euler characteristic 1 |

It has 6 pentagonal faces, 15 edges, and 10 vertices.

## Projections

It can be projected symmetrically inside of a 10-sided or 12-sided perimeter:

## Petersen graph

From the point of view of graph theory this is an embedding of Petersen graph on a real projective plane.
With this embedding, the dual graph is
*K*_{6} (the complete graph with 6 vertices) --- see hemi-icosahedron.

## See also

- 57-cell – an abstract regular 4-polytope constructed from 57 hemi-dodecahedra.
- hemi-icosahedron
- hemi-cube
- hemi-octahedron

## References

- McMullen, Peter; Schulte, Egon (December 2002), "6C. Projective Regular Polytopes",
*Abstract Regular Polytopes*(1st ed.), Cambridge University Press, pp. 162–165, ISBN 0-521-81496-0

## External links

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