# Hemi-octahedron

A **hemi-octahedron** is an abstract regular polyhedron, containing half the faces of a regular octahedron.

Hemi-octahedron | |
---|---|

Type | abstract regular polyhedron globally projective polyhedron |

Faces | 4 triangles |

Edges | 6 |

Vertices | 3 |

Vertex configuration | 3.3.3.3 |

Schläfli symbol | {3,4}/2 or {3,4}_{3} |

Symmetry group | S_{4}, order 24 |

Dual polyhedron | hemicube |

Properties | non-orientable Euler characteristic 1 |

It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the hemicube.

It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 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 four equal parts. It can be seen as a square pyramid without its base.

It can be represented symmetrically as a hexagonal or square Schlegel diagram:

It has an unexpected property that there are two distinct edges between every pair of vertices – any two vertices define a digon.

## See also

## 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

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