Cantellated 120-cell

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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-cell[edit]

Cantellated 120-cell
Type Uniform 4-polytope
Uniform index 37
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 1920 total:
120 (3.4.5.4) Small rhombicosidodecahedron.png
1200 (3.4.4) Triangular prism.png
600 (3.3.3.3) Octahedron.png
Faces 4800{3}+3600{4}+720{5}
Edges 10800
Vertices 3600
Vertex figure Cantellated 120-cell verf.png
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 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
120-cell t02 H3.png
[10]
120-cell t02 B3.png
[6]
120-cell t02 A3.png
[4]
Cantellated 120 cell center.png
Schlegel diagram. Pentagonal face are removed.

Cantitruncated 120-cell[edit]

Cantitruncated 120-cell
Type Uniform 4-polytope
Uniform index 42
Schläfli symbol t0,1,2{5,3,3}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 1920 total:
120 (4.6.10) Great rhombicosidodecahedron.png
1200 (3.4.4) Triangular prism.png
600 (3.6.6) Truncated tetrahedron.png
Faces 9120:
2400{3}+3600{4}+
2400{6}+720{10}
Edges 14400
Vertices 7200
Vertex figure Cantitruncated 120-cell verf.png
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 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
120-cell t012 H3.png
[10]
120-cell t012 B3.png
[6]
120-cell t012 A3.png
[4]
Schlegel diagram
Cantitruncated 120-cell.png
Centered on truncated icosidodecahedron cell with decagonal faces hidden.

Cantellated 600-cell[edit]

Cantellated 600-cell
Type Uniform 4-polytope
Uniform index 40
Schläfli symbol t0,2{3,3,5}
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 1440 total:
120 Icosidodecahedron.png 3.5.3.5
600 Cuboctahedron.png 3.4.3.4
720 Pentagonal prism.png 4.4.5
Faces 8640 total:
(1200+2400){3}
+3600{4}+1440{5}
Edges 10800
Vertices 3600
Vertex figure Cantellated 600-cell verf.png
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 names[edit]

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
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cell Picture
0 600 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Cantellated tetrahedron
(Cuboctahedron)
Cantellated tetrahedron.png
1 1200 CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png None
(Degenerate triangular prism)
 
2 720 CDel node.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png Pentagonal prism Pentagonal prism.png
3 120 CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png Rectified dodecahedron
(Icosidodecahedron)
Icosidodecahedron.png

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 -
600-cell t02 H4.svg
[30]
600-cell t02 p20.svg
[20]
F4 H3
600-cell t02 F4.svg
[12]
600-cell t02 H3.svg
[10]
A2 / B3 / D4 A3 / B2
600-cell t02 B3.svg
[6]
600-cell t02 B2.svg
[4]
Schlegel diagrams
Cantitruncated 600-cell.png Cantellated 600 cell center.png
Stereographic projection with its 3600 green triangular faces and its 3600 blue square faces.

Cantitruncated 600-cell[edit]

Cantitruncated 600-cell
Type Uniform 4-polytope
Uniform index 45
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells 1440 total:
120 (5.6.6) Truncated icosahedron.png
720 (4.4.5) Pentagonal prism.png
600 (4.6.6) Truncated octahedron.png
Faces 8640:
3600{4}+1440{5}+
3600{6}
Edges 14400
Vertices 7200
Vertex figure Cantitruncated 600-cell verf.png
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 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
Cantitruncated 600-cell.png
Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2
120-cell t123 H3.png
[10]
120-cell t123 B3.png
[6]
120-cell t123 A3.png
[4]

Related polytopes[edit]

Notes[edit]

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

References[edit]

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds