# Hemipolytope

**Hemipolytopes** are polytopes with facets that pass through the center of the polytope. They were first studied in the context of uniform polyhedra, where there are ten **hemipolyhedra**, but the term may apply to any polytope that can be inscribed in a hypersphere, such as isogonal and orbiform polytopes. Many, but not all, hemipolytopes are non-orientable.

The "hemi-" prefix refers to a property of the uniform hemipolyhedra (except the great dirhombicosidodecahedron) where the facets that pass through the center, called **hemi facets**, are parallel to the faces of a regular polyhedron and number half the faces of that polyhedron. For example, the octahemioctahedron's hemi faces comprise four regular hexagons, each parallel to two of the eight triangles of a regular octahedron.

Hemipolytopes are often derived as facetings of other polytopes. All polytopes in the demicross series are uniform hemipolytopes, but not all hemipolytopes are demicrosses.

There are no uniform hemipolygons, but there are infinitely many isogonal polygons with edges that pass through the center, such as the bowtie and the propeller tripod with edge lengths 2:1.

## List of uniform hemipolyhedra[edit | edit source]

There are 10 uniform polyhedra which are hemipolyhedra. All of them except the octahemioctahedron and the great dirhombicosidodecahedron are non-orientable. Additionally, all are edge-transitive and two-orbit except the great dirhombicosidodecahedron.

Name | Short
Name |
Image | Normal Faces | Hemi Faces | Regiment |
---|---|---|---|---|---|

Tetrahemihexahedron | thah | 4 triangles | 3 squares | Octahedron | |

Octahemioctahedron | oho | 8 triangles | 4 hexagons | Cuboctahedron | |

Cubohemioctahedron | cho | 6 squares | |||

Small dodecahemidodecahedron | sidhid | 12 pentagons | 6 decagons | Icosidodecahedron | |

Small icosihemidodecahedron | seihid | 20 triangles | |||

Great dodecahemicosahedron | gidhei | 12 pentagons | 10 hexagons | Dodecadodecahedron | |

Small dodecahemicosahedron | sidhei | 12 pentagrams | |||

Great icosihemidodecahedron | geihid | 20 triangles | 6 decagrams | Great icosidodecahedron | |

Great dodecahemidodecahedron | gidhid | 12 pentagrams | |||

Great dirhombicosidodecahedron | gidrid | 40 triangles
24 pentagrams |
60 squares | Great dirhombicosidodecahedron |

## Other hemipolytopes[edit | edit source]

As previously mentioned, all polytopes in the demicross series are hemipolytopes. Likewise, a prism of a hemipolytope will result in another hemipolytope. A hemipolytope pyramid or tegum will result in another hemipolytope if the center of the original polytope's facets are the center of the polytope.

Here is an incomplete list of hemipolytopes.

Rank | Name | Short
Name |
Image | Normal Facets | Hemi Facets | Properties | Army /
Regiment |
---|---|---|---|---|---|---|---|

2 | Bowtie /
2D Demicross |
bowtie | 2 dyads | 2 dyads | Semi-uniform | Rectangle | |

2 | Centered Tripod | 3 dyads | 3 dyads | Semi-uniform | Hexagon | ||

2 | Centered Pentapod | 5 dyads | 5 dyads | Semi-uniform | Decagon | ||

3 | Bowtie Prism | 2 bowties
2 squares |
2 rectangles | Semi-uniform | Cube | ||

3 | Bowtie Tegum | bobipyr | 4 triangles | 2 squares | Orbiform | Octahedron | |

4 | Tesseractihemioctachoron/ 4D Demicross | tho | 8 tetrahedra | 4 octahedra | Uniform | Hexadecachoron | |

4 | Great ditrigonary icosidodecahedral antiprism | gidtidap | 2 great ditrigonary icosidodecahedra
40 tetrahedra |
12 pentagonal antiprisms | Uniform | Sidtidap | |

5 | Hexadecahemidecateron / 5D Demicross | hehad | 16 pentachora | 5 hexadecachora | Uniform | Triacontaditeron |