Hemipolytope

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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