# Hemi-icosahedron

A **hemi-icosahedron** is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 10 triangles), 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-icosahedron | |
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decagonal Schlegel diagram | |

Type | abstract regular polyhedron globally projective polyhedron |

Faces | 10 triangles |

Edges | 15 |

Vertices | 6 |

Vertex configuration | 3.3.3.3.3 |

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

Symmetry group | A_{5}, order 60 |

Dual polyhedron | hemi-dodecahedron |

Properties | non-orientable Euler characteristic 1 |

## Geometry

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

It is also related to the nonconvex uniform polyhedron, the tetrahemihexahedron, which could be topologically identical to the hemi-icosahedron if each of the 3 square faces were divided into two triangles.

## Graphs

It can be represented symmetrically on faces, and vertices as schlegel diagrams:

Face-centered |
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## The complete graph K6

It has the same vertices and edges as the 5-dimensional 5-simplex which has a complete graph of edges, but only contains half of the (20) faces.

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

## See also

- 11-cell - an abstract regular 4-polytope constructed from 11 hemi-icosahedra.
- hemi-dodecahedron
- 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