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Four cantellations
120-cell t0 H3.svg
120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
120-cell t02 H3.png
Cantellated 120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
600-cell t02 H3.svg
Cantellated 600-cell
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
600-cell t0 H3.svg
600-cell
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
120-cell t012 H3.png
Cantitruncated 120-cell
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
120-cell t123 H3.png
Cantitruncated 600-cell
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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-cellEdit

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

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 namesEdit

  • 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)

ImagesEdit

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

Cantitruncated 120-cellEdit

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

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 namesEdit

  • 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)

ImagesEdit

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-cellEdit

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

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 namesEdit

ConstructionEdit

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.

ImagesEdit

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-cellEdit

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

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 namesEdit

  • 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)

ImagesEdit

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

Related polytopesEdit

NotesEdit

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

ReferencesEdit

  • Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 37, 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
  • 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
  • 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 - grix

External linksEdit