Cantellated 120-cell

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.

Four cantellations

120-cell

Cantellated 120-cell

Cantellated 600-cell

600-cell

Cantitruncated 120-cell

Cantitruncated 600-cell
Orthogonal projections in H3 Coxeter plane

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
TypeUniform 4-polytope
Uniform index37
Coxeter diagram
Cells1920 total:
120 (3.4.5.4)
1200 (3.4.4)
600 (3.3.3.3)
Faces4800{3}+3600{4}+720{5}
Edges10800
Vertices3600
Vertex figure
wedge
Schläfli symbolt0,2{5,3,3}
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex

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
TypeUniform 4-polytope
Uniform index42
Schläfli symbolt0,1,2{5,3,3}
Coxeter diagram
Cells1920 total:
120 (4.6.10)
1200 (3.4.4)
600 (3.6.6)
Faces9120:
2400{3}+3600{4}+
2400{6}+720{10}
Edges14400
Vertices7200
Vertex figure
sphenoid
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex

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
TypeUniform 4-polytope
Uniform index40
Schläfli symbolt0,2{3,3,5}
Coxeter diagram
Cells1440 total:
120 3.5.3.5
600 3.4.3.4
720 4.4.5
Faces8640 total:
(1200+2400){3}
+3600{4}+1440{5}
Edges10800
Vertices3600
Vertex figure
isosceles triangular prism
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex

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
TypeUniform 4-polytope
Uniform index45
Coxeter diagram
Cells1440 total:
120 (5.6.6)
720 (4.4.5)
600 (4.6.6)
Faces8640:
3600{4}+1440{5}+
3600{6}
Edges14400
Vertices7200
Vertex figure
sphenoid
Schläfli symbolt0,1,2{3,3,5}
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex

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

Schlegel diagram
Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2

[10]

[6]

[4]

Notes

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

References

  • 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 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
Family An Bn I2(p) / Dn E6 / E7 / E8 / 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
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