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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.

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[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 isotoxal (edge-transitive) except the great dirhombicosidodecahedron.

Name Short


Image Normal Faces Hemi Faces Regiment
Tetrahemihexahedron thah Tetrahemihexahedron.png 4 triangles 3 squares

Octahedron equator.png

Octahemioctahedron oho Octahemioctahedron.png 8 triangles 4 hexagons

Cuboctahedron equator.png

Cubohemioctahedron cho Cubohemioctahedron.png 6 squares
Small dodecahemidodecahedron sidhid Small dodecahemidodecahedron.png 12 pentagons 6 decagons

Icosidodecahedron equator.png

Small icosihemidodecahedron seihid Small icosihemidodecahedron.png 20 triangles
Great dodecahemicosahedron gidhei Great dodecahemicosahedron.png 12 pentagons 10 hexagons

Dodecadodecahedron equator.png

Small dodecahemicosahedron sidhei Small dodecahemicosahedron.png 12 pentagrams
Great icosihemidodecahedron geihid Great icosihemidodecahedron.png 20 triangles 6 decagrams

Great icosidodecahedron equator.png

Great icosidodecahedron
Great dodecahemidodecahedron gidhid Great dodecahemidodecahedron.png 12 pentagrams
Great dirhombicosidodecahedron gidrid Great dirhombicosidodecahedron.png 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


Image Normal Facets Hemi Facets Properties Army /


2 Bowtie /

2D Demicross

bowtie Bowtie.svg 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 Bowtie prism.png 2 bowties

2 squares

2 rectangles Semi-uniform Cube
3 Bowtie Tegum bobipyr Bowtie tegum.png 4 triangles 2 squares Orbiform Octahedron
4 Tesseractihemioctachoron/ 4D Demicross tho Tho.png 8 tetrahedra 4 octahedra Uniform Hexadecachoron
4 Great ditrigonary icosidodecahedral antiprism gidtidap Gidtidap.png 2 great ditrigonary icosidodecahedra

40 tetrahedra

12 pentagonal antiprisms Uniform Sidtidap
5 Hexadecahemidecateron / 5D Demicross hehad Hehad.PNG 16 pentachora 5 hexadecachora Uniform Triacontaditeron