# 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.

While there are no uniform hemipolygons, the infinite series of centered polypods are semi-uniform. The bowtie, a faceted rectangle, is a similar case outside of this infinite family.

## List of Uniform Hemipolyhedra

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

## Other Hemipolytopes

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
3 Bowtie Prism 2 bowties 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 12 pentagonal antiprisms Uniform Sidtidap
5 Hexadecahemidecateron / 5D Demicross hehad 16 pentachora 5 hexadecachora Uniform Triacontaditeron