Cantellated 120-cell

File:Small rhombated hecatonicosachoron net.png]]

The cantellated 120-cell is a uniform 4-polytope. It is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra, 1200 triangular prisms, 600 octahedra. Its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex.

• Cantellated 120-cell Norman Johnson
• Cantellated hecatonicosachoron / Cantellated dodecacontachoron / Cantellated polydodecahedron
• Small rhombated hecatonicosachoron (Acronym {{not a typo|srahi}}) (George Olshevsky and Jonathan Bowers)Klitzing, (o3x3o5x - srahi)
• Ambo-02 polydodecahedron (John Conway)

File:Great rhombated hecatonicosachoron net.png]]

The cantitruncated 120-cell is a uniform polychoron.

This 4-polytope is related to the regular 120-cell. The cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells.

The image shows the 4-polytope drawn as a Schlegel diagram which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the decagonal faces are hidden, allowing us to see the elemented projected inside.

• Cantitruncated 120-cell Norman Johnson
• Cantitruncated hecatonicosachoron / Cantitruncated polydodecahedron
• Great rhombated hecatonicosachoron (Acronym grahi) (George Olshevsky and Jonthan Bowers)Klitzing, (o3x3x5x - grahi)
• Ambo-012 polydodecahedron (John Conway)

File:Small rhombated hexacosichoron net.png]]

The cantellated 600-cell is a uniform 4-polytope. It has 1440 cells: 120 icosidodecahedra, 600 cuboctahedra, and 720 pentagonal prisms. Its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms.

• Cantellated 600-cell Norman Johnson
• Cantellated hexacosichoron / Cantellated tetraplex
• Small {{not a typo|rhombihexacosichoron}} (Acronym srix) (George Olshevsky and Jonathan Bowers)Klitzing, (x3o3x5o - srix)
• Ambo-02 tetraplex (John Conway)

This 4-polytope has cells at 3 of 4 positions in the fundamental domain, extracted from the Coxeter diagram by removing one node at a time:

There are 1440 pentagonal faces between the icosidodecahedra and pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms. There are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra.

There are two classes of edges: 3-4-4, 3-4-5: 3600 have two squares and a triangle around it, and 7200 have one triangle, one square, and one pentagon.

File:Great rhombated hexacosichoron net.png]]

The cantitruncated 600-cell is a uniform 4-polytope. It is composed of 1440 cells: 120 truncated icosahedra, 720 pentagonal prisms and 600 truncated octahedra. It has 7200 vertices, 14400 edges, and 8640 faces (3600 squares, 1440 pentagons, and 3600 hexagons). It has an irregular tetrahedral vertex figure, filled by one truncated icosahedron, one pentagonal prism and two truncated octahedra.

• Cantitruncated 600-cell (Norman Johnson)
• Cantitruncated hexacosichoron / Cantitruncated polydodecahedron
• Great rhombated hexacosichoron (acronym grix) (George Olshevsky and Jonathan Bowers)Klitzing, (x3x3x5o - grix)
• Ambo-012 polytetrahedron (John Conway)

{{H4_family}}

{{reflist}}

• {{PolyCell | urlname = section4.html| title = Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 37}}
• [http://www.polytope.de/nr57.html Archimedisches Polychor Nr. 57 (cantellated 120-cell)] Marco Möller's Archimedean polytopes in R⁴ (German)
• [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Kaleidoscopes: Selected Writings of H.S.M. Coxeter], edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}}
• (Paper 22) H.S.M. Coxeter, [https://doi.org/10.1007%2FBF01181449 Regular and Semi Regular Polytopes I], [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, [https://doi.org/10.1007%2FBF01161657 Regular and Semi-Regular Polytopes II], [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, [https://doi.org/10.1007%2FBF01161745 Regular and Semi-Regular Polytopes III], [Math. Zeit. 200 (1988) 3-45]
• J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• [http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }} [http://www.polytope.de/nr63.html m63] [http://www.polytope.de/nr61.html m61] [http://www.polytope.de/nr56.html m56]
• {{PolyCell | urlname = section4.html| title = Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 40, 42, 45}}
• {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} o3x3o5x - srahi, o3x3x5x - grahi, x3o3x5o - srix, x3x3x5o - grix

• [http://www.georgehart.com/zome-polytopes-ISAMA07/hart-isama07.doc Four-Dimensional Polytope Projection Barn Raisings] (A Zometool construction of the cantitruncated 120-cell), George W. Hart
• [http://homepages.wmich.edu/~drichter/bridgeszome2005.htm Renaissance Banff 2005 Zome Project]: a Zome model of a 3D orthogonal projection of the cantellated 600-cell.
• H4 uniform polytopes with coordinates: [http://eusebeia.dyndns.org/4d/cant600cell rr{3,3,5}] [http://eusebeia.dyndns.org/4d/cant120cell rr{5,3,3}] [http://eusebeia.dyndns.org/4d/cantitrunc600cell tr{3,3,5}] [http://eusebeia.dyndns.org/4d/cantitrunc120cell tr{5,3,3}]

{{Polytopes}}

Category:Uniform 4-polytopes