# Octahemioctahedron

Octahemioctahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Oho |

Coxeter diagram | x3/2o3x3*a () |

Elements | |

Faces | 8 triangles, 4 hexagons |

Edges | 24 |

Vertices | 12 |

Vertex figure | Bowtie, edge lengths 1 and √3 |

Measures (edge length 1) | |

Circumradius | 1 |

Volume | 0 |

Dihedral angle | |

Central density | 0 |

Number of external pieces | 32 |

Level of complexity | 4 |

Related polytopes | |

Army | Co |

Regiment | Co |

Dual | Octahemioctacron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 96 |

Euler characteristic | 0 |

Orientable | Yes |

Genus | 1 |

Properties | |

Symmetry | B_{3}, order 48 |

Flag orbits | 2 |

Convex | No |

Nature | Tame |

The **octahemioctahedron**, or **oho**, is a quasiregular polyhedron and one of 10 uniform hemipolyhedra. It consists of 8 equilateral triangles and 4 "hemi" hexagons passing through its center, with two of each joining at a vertex. It also has 6 square pseudofaces. Its triangular faces, as well as its hemi hexagonal faces, are parallel to those of an octahedron: hence the name. It can be derived as a rectified petrial cube.

Unlike the other quasiregular hemipolyhedra, the octahemioctahedron is orientable. This is because the triangles can be grouped into two sets of four, one of which is seen as positive-density and the other seen as negative-density. In addition, while it has cubic symmetry, its Coxeter diagram also makes it uniform under tetrahedral symmetry.

The visible portion of this solid resembles a cuboctahedron with six square pyramids carved out. In fact the triangular faces are the same ones as from the cuboctahedron, while the hexagons are those of the cuboctahedron's equatorial planes.

The octahemioctahedron is topologically a torus, and is isomorphic to a quotient of the trihexagonal tiling.

## Vertex coordinates[edit | edit source]

Its vertices are the same as those of its regiment colonel, the cuboctahedron.

## Representations[edit | edit source]

An octahemioctahedron has the following Coxeter diagrams:

- x3/2o3x3*a
- ß3o3x
- ß3/2x3x

## Related polyhedra[edit | edit source]

The icosidisicosahedron is a uniform polyhedron compound composed of 5 octahemioctahedra.

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#22).

- Bowers, Jonathan. "Batch 1: Oct and Co Facetings" (#2 under co).

- Klitzing, Richard. "oho".
- Wikipedia contributors. "Octahemioctahedron".
- McCooey, David. "Octahemioctahedron"