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Cantellated 120-cell

August 17, 2023

Four cantellations

120-cell
Cantellated 120-cell
Cantellated 600-cell

600-cell
Cantitruncated 120-cell
Cantitruncated 600-cell
Orthogonal projections in H3 Coxeter plane

In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 120-cell.

There are four degrees of cantellations of the 120-cell including with permutations truncations. Two are expressed relative to the dual 600-cell.

Cantellated 120-cell

Cantellated 120-cell
Type Uniform 4-polytope
Uniform index 37
Coxeter diagram        
Cells 1920 total:
120 (3.4.5.4)  
1200 (3.4.4)  
600 (3.3.3.3)  
Faces 4800{3}+3600{4}+720{5}
Edges 10800
Vertices 3600
Vertex figure  
wedge
Schläfli symbol t0,2{5,3,3}
Symmetry group H4, [3,3,5], order 14400
Properties convex
 
Net

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.

Alternative names

  • Cantellated 120-cell Norman Johnson
  • Cantellated hecatonicosachoron / Cantellated dodecacontachoron / Cantellated polydodecahedron
  • Small rhombated hecatonicosachoron (Acronym srahi) (George Olshevsky and Jonathan Bowers)[1]
  • Ambo-02 polydodecahedron (John Conway)

Images

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2
 
[10] 		 
[6] 		 
[4]
 
Schlegel diagram. Pentagonal face are removed.

Cantitruncated 120-cell

Cantitruncated 120-cell
Type Uniform 4-polytope
Uniform index 42
Schläfli symbol t0,1,2{5,3,3}
Coxeter diagram        
Cells 1920 total:
120 (4.6.10)  
1200 (3.4.4)  
600 (3.6.6)  
Faces 9120:
2400{3}+3600{4}+
2400{6}+720{10}
Edges 14400
Vertices 7200
Vertex figure  
sphenoid
Symmetry group H4, [3,3,5], order 14400
Properties convex
 
Net

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.

Alternative names

  • Cantitruncated 120-cell Norman Johnson
  • Cantitruncated hecatonicosachoron / Cantitruncated polydodecahedron
  • Great rhombated hecatonicosachoron (Acronym grahi) (George Olshevsky and Jonthan Bowers)[2]
  • Ambo-012 polydodecahedron (John Conway)

Images

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2
 
[10] 		 
[6] 		 
[4]
Schlegel diagram
 
Centered on truncated icosidodecahedron cell with decagonal faces hidden.

Cantellated 600-cell

Cantellated 600-cell
Type Uniform 4-polytope
Uniform index 40
Schläfli symbol t0,2{3,3,5}
Coxeter diagram        
Cells 1440 total:
120   3.5.3.5
600   3.4.3.4
720   4.4.5
Faces 8640 total:
(1200+2400){3}
+3600{4}+1440{5}
Edges 10800
Vertices 3600
Vertex figure  
isosceles triangular prism
Symmetry group H4, [3,3,5], order 14400
Properties convex
 
Net

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.

Alternative names

  • Cantellated 600-cell Norman Johnson
  • Cantellated hexacosichoron / Cantellated tetraplex
  • Small rhombihexacosichoron (Acronym srix) (George Olshevsky and Jonathan Bowers)[3]
  • Ambo-02 tetraplex (John Conway)

Construction

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:

Node Order Coxeter diagram
        		Cell Picture
0 600       Cantellated tetrahedron
(Cuboctahedron) 		 
1 1200       None
(Degenerate triangular prism) 		 
2 720       Pentagonal prism  
3 120       Rectified dodecahedron
(Icosidodecahedron) 		 

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.

Images

Orthographic projections by Coxeter planes
H4 -
 
[30] 		 
[20]
F4 H3
 
[12] 		 
[10]
A2 / B3 / D4 A3 / B2
 
[6] 		 
[4]
Schlegel diagrams
   
Stereographic projection with its 3600 green triangular faces and its 3600 blue square faces.

Cantitruncated 600-cell

Cantitruncated 600-cell
Type Uniform 4-polytope
Uniform index 45
Coxeter diagram        
Cells 1440 total:
120 (5.6.6)  
720 (4.4.5)  
600 (4.6.6)  
Faces 8640:
3600{4}+1440{5}+
3600{6}
Edges 14400
Vertices 7200
Vertex figure  
sphenoid
Schläfli symbol t0,1,2{3,3,5}
Symmetry group H4, [3,3,5], order 14400
Properties convex
 
Net

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.

Alternative names

  • Cantitruncated 600-cell (Norman Johnson)
  • Cantitruncated hexacosichoron / Cantitruncated polydodecahedron
  • Great rhombated hexacosichoron (acronym grix) (George Olshevsky and Jonathan Bowers)[4]
  • Ambo-012 polytetrahedron (John Conway)

Images

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2
 
[10] 		 
[6] 		 
[4]

Related polytopes

H4 family polytopes
120-cell rectified
120-cell 		truncated
120-cell 		cantellated
120-cell 		runcinated
120-cell 		cantitruncated
120-cell 		runcitruncated
120-cell 		omnitruncated
120-cell
                                                               
{5,3,3} r{5,3,3} t{5,3,3} rr{5,3,3} t0,3{5,3,3} tr{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
               
             
600-cell rectified
600-cell 		truncated
600-cell 		cantellated
600-cell 		bitruncated
600-cell 		cantitruncated
600-cell 		runcitruncated
600-cell 		omnitruncated
600-cell
                                                               
{3,3,5} r{3,3,5} t{3,3,5} rr{3,3,5} 2t{3,3,5} tr{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}

Notes

  1. ^ Klitzing, (o3x3o5x - srahi)
  2. ^ Klitzing, (o3x3x5x - grahi)
  3. ^ Klitzing, (x3o3x5o - srix)
  4. ^ Klitzing, (x3x3x5o - grix)

References

  • , George Olshevsky.
  • Archimedisches Polychor Nr. 57 (cantellated 120-cell) Marco Möller's Archimedean polytopes in R4 (German)
  • 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, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, 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
  • Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [1] m63 m61 m56
  • , George Olshevsky.
  • Klitzing, Richard. "4D uniform polytopes (polychora)". o3x3o5x - srahi, o3x3x5x - grahi, x3o3x5o - srix, x3x3x5o - grix

External links

  • Four-Dimensional Polytope Projection Barn Raisings (A Zometool construction of the cantitruncated 120-cell), George W. Hart
  • Renaissance Banff 2005 Zome Project: a Zome model of a 3D orthogonal projection of the cantellated 600-cell.
  • H4 uniform polytopes with coordinates: rr{3,3,5} rr{5,3,3} tr{3,3,5} tr{5,3,3}
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

August 17, 2023
cantellated, cell, four, cantellations, cell, cantellated, cell600, cell, cantitruncated, cell, cantitruncated, cellorthogonal, projections, coxeter, planein, four, dimensional, geometry, cantellated, cell, convex, uniform, polytope, being, cantellation, order. Four cantellations 120 cell Cantellated 120 cell Cantellated 600 cell600 cell Cantitruncated 120 cell Cantitruncated 600 cellOrthogonal projections in H3 Coxeter planeIn four dimensional geometry a cantellated 120 cell is a convex uniform 4 polytope being a cantellation a 2nd order truncation of the regular 120 cell There are four degrees of cantellations of the 120 cell including with permutations truncations Two are expressed relative to the dual 600 cell Contents 1 Cantellated 120 cell 1 1 Alternative names 1 2 Images 2 Cantitruncated 120 cell 2 1 Alternative names 2 2 Images 3 Cantellated 600 cell 3 1 Alternative names 3 2 Construction 3 3 Images 4 Cantitruncated 600 cell 4 1 Alternative names 4 2 Images 5 Related polytopes 6 Notes 7 References 8 External linksCantellated 120 cell EditCantellated 120 cellType Uniform 4 polytopeUniform index 37Coxeter diagram Cells 1920 total 120 3 4 5 4 1200 3 4 4 600 3 3 3 3 Faces 4800 3 3600 4 720 5 Edges 10800Vertices 3600Vertex figure wedgeSchlafli symbol t0 2 5 3 3 Symmetry group H4 3 3 5 order 14400Properties convex NetThe 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 Alternative names Edit Cantellated 120 cell Norman Johnson Cantellated hecatonicosachoron Cantellated dodecacontachoron Cantellated polydodecahedron Small rhombated hecatonicosachoron Acronym srahi George Olshevsky and Jonathan Bowers 1 Ambo 02 polydodecahedron John Conway Images Edit Orthographic projections by Coxeter planes H3 A2 B3 D4 A3 B2 10 6 4 Schlegel diagram Pentagonal face are removed Cantitruncated 120 cell EditCantitruncated 120 cellType Uniform 4 polytopeUniform index 42Schlafli symbol t0 1 2 5 3 3 Coxeter diagram Cells 1920 total 120 4 6 10 1200 3 4 4 600 3 6 6 Faces 9120 2400 3 3600 4 2400 6 720 10 Edges 14400Vertices 7200Vertex figure sphenoidSymmetry group H4 3 3 5 order 14400Properties convex NetThe 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 Alternative names Edit Cantitruncated 120 cell Norman Johnson Cantitruncated hecatonicosachoron Cantitruncated polydodecahedron Great rhombated hecatonicosachoron Acronym grahi George Olshevsky and Jonthan Bowers 2 Ambo 012 polydodecahedron John Conway Images Edit Orthographic projections by Coxeter planes H3 A2 B3 D4 A3 B2 10 6 4 Schlegel diagram Centered on truncated icosidodecahedron cell with decagonal faces hidden Cantellated 600 cell EditCantellated 600 cellType Uniform 4 polytopeUniform index 40Schlafli symbol t0 2 3 3 5 Coxeter diagram Cells 1440 total 120 3 5 3 5600 3 4 3 4720 4 4 5Faces 8640 total 1200 2400 3 3600 4 1440 5 Edges 10800Vertices 3600Vertex figure isosceles triangular prismSymmetry group H4 3 3 5 order 14400Properties convex NetThe 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 Alternative names Edit Cantellated 600 cell Norman Johnson Cantellated hexacosichoron Cantellated tetraplex Small rhombihexacosichoron Acronym srix George Olshevsky and Jonathan Bowers 3 Ambo 02 tetraplex John Conway Construction Edit 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 Node Order Coxeter diagram Cell Picture0 600 Cantellated tetrahedron Cuboctahedron 1 1200 None Degenerate triangular prism 2 720 Pentagonal prism 3 120 Rectified dodecahedron Icosidodecahedron 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 Images Edit Orthographic projections by Coxeter planes H4 30 20 F4 H3 12 10 A2 B3 D4 A3 B2 6 4 Schlegel diagrams Stereographic projection with its 3600 green triangular faces and its 3600 blue square faces Cantitruncated 600 cell EditCantitruncated 600 cellType Uniform 4 polytopeUniform index 45Coxeter diagram Cells 1440 total 120 5 6 6 720 4 4 5 600 4 6 6 Faces 8640 3600 4 1440 5 3600 6 Edges 14400Vertices 7200Vertex figure sphenoidSchlafli symbol t0 1 2 3 3 5 Symmetry group H4 3 3 5 order 14400Properties convex NetThe 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 Alternative names Edit Cantitruncated 600 cell Norman Johnson Cantitruncated hexacosichoron Cantitruncated polydodecahedron Great rhombated hexacosichoron acronym grix George Olshevsky and Jonathan Bowers 4 Ambo 012 polytetrahedron John Conway Images Edit Schlegel diagram Orthographic projections by Coxeter planes H3 A2 B3 D4 A3 B2 10 6 4 Related polytopes EditH4 family polytopes120 cell rectified120 cell truncated120 cell cantellated120 cell runcinated120 cell cantitruncated120 cell runcitruncated120 cell omnitruncated120 cell 5 3 3 r 5 3 3 t 5 3 3 rr 5 3 3 t0 3 5 3 3 tr 5 3 3 t0 1 3 5 3 3 t0 1 2 3 5 3 3 600 cell rectified600 cell truncated600 cell cantellated600 cell bitruncated600 cell cantitruncated600 cell runcitruncated600 cell omnitruncated600 cell 3 3 5 r 3 3 5 t 3 3 5 rr 3 3 5 2t 3 3 5 tr 3 3 5 t0 1 3 3 3 5 t0 1 2 3 3 3 5 Notes Edit Klitzing o3x3o5x srahi Klitzing o3x3x5x grahi Klitzing x3o3x5o srix Klitzing x3x3x5o grix References EditConvex uniform polychora based on the hecatonicosachoron 120 cell and hexacosichoron 600 cell Model 37 George Olshevsky Archimedisches Polychor Nr 57 cantellated 120 cell Marco Moller s Archimedean polytopes in R4 German 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 Regular and Semi Regular Polytopes I Math Zeit 46 1940 380 407 MR 2 10 Paper 23 H S M Coxeter Regular and Semi Regular Polytopes II Math Zeit 188 1985 559 591 Paper 24 H S M Coxeter 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 Four dimensional Archimedean Polytopes German Marco Moller 2004 PhD dissertation 1 m63 m61 m56 Convex uniform polychora based on the hecatonicosachoron 120 cell and hexacosichoron 600 cell Model 40 42 45 George Olshevsky Klitzing Richard 4D uniform polytopes polychora o3x3o5x srahi o3x3x5x grahi x3o3x5o srix x3x3x5o grixExternal links EditFour Dimensional Polytope Projection Barn Raisings A Zometool construction of the cantitruncated 120 cell George W Hart Renaissance Banff 2005 Zome Project a Zome model of a 3D orthogonal projection of the cantellated 600 cell H4 uniform polytopes with coordinates rr 3 3 5 rr 5 3 3 tr 3 3 5 tr 5 3 3 vteFundamental convex regular and uniform polytopes in dimensions 2 10Family An Bn I2 p Dn E6 E7 E8 F4 G2 HnRegular polygon Triangle Square p gon Hexagon PentagonUniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron IcosahedronUniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cellUniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicubeUniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicubeUniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicubeUniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytopeTopics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Cantellated 120 cell amp oldid 1148108279, wikipedia, wiki, book, books, library,

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