Hemipolytope

This article is about polytopes with facets that pass through their centers. For polytopes with opposite vertices identified, see Projective polytope.

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.

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

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