Runcinated 120-cells

Four runcinations

120-cell
Runcinated 120-cell
(Expanded 120-cell)
Runcitruncated 120-cell

600-cell
Runcitruncated 600-cell
Omnitruncated 120-cell
Orthogonal projections in H3 Coxeter plane

In four-dimensional geometry, a runcinated 120-cell (or runcinated 600-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 120-cell.

There are 4 degrees of runcinations of the 120-cell including with permutations truncations and cantellations.

The runcinated 120-cell can be seen as an expansion applied to a regular 4-polytope, the 120-cell or 600-cell.

Runcinated 120-cell

Runcinated 120-cell
TypeUniform 4-polytope
Uniform index38
Coxeter diagram
Cells2640 total:
120 5.5.5
720 4.4.5
1200 4.4.3
600 3.3.3
Faces7440:
2400{3}+3600{4}+
1440{5}
Edges7200
Vertices2400
Vertex figure
Equilateral-triangular antipodium
Schläfli symbol t0,3{5,3,3}
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex
Net

The runcinated 120-cell or small disprismatohexacosihecatonicosachoron is a uniform 4-polytope. It has 2640 cells: 120 dodecahedra, 720 pentagonal prisms, 1200 triangular prisms, and 600 tetrahedra. Its vertex figure is a nonuniform triangular antiprism (equilateral-triangular antipodium): its bases represent a dodecahedron and a tetrahedron, and its flanks represent three triangular prisms and three pentagonal prisms.

Alternate names

Images

Schlegel diagram (Only tetrahedral cells shown)
Orthogonal projections in Coxeter planes

H3
A2/B3
A3/B2

Runcitruncated 120-cell

Runcitruncated 120-cell
TypeUniform 4-polytope
Uniform index43
Coxeter diagram
Cells2640 total:
120 (3.10.10)

720 (4.4.10)
1200 (3.4.4)
600 (3.4.3.4)

Faces13440:
4800{3}+7200{4}+
1440{10}
Edges18000
Vertices7200
Vertex figure
Irregular rectangular pyramid
Schläfli symbol t0,1,3{5,3,3}
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex
Net

The runcitruncated 120-cell or prismatorhombated hexacosichoron is a uniform 4-polytope. It contains 2640 cells: 120 truncated dodecahedra, 720 decagonal prisms, 1200 triangular prisms, and 600 cuboctahedra. Its vertex figure is an irregular rectangular pyramid, with one truncated dodecahedron, two decagonal prisms, one triangular prism, and one cuboctahedron.

Alternate names

Images

Schlegel diagram (Only triangular prisms shown)
Orthogonal projections in Coxeter planes

H3
A2/B3
A3/B2

Runcitruncated 600-cell

Runcitruncated 600-cell
TypeUniform 4-polytope
Uniform index44
Coxeter diagram
Cells2640 total:
120 3.4.5.4
720 4.4.5
1200 4.4.6
600 3.6.6
Faces13440:
2400{3}+7200{4}+
1440{5}+2400{6}
Edges18000
Vertices7200
Vertex figure
Trapezoidal pyramid
Schläfli symbol t0,1,3{3,3,5}
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex
Net

The runcitruncated 600-cell or prismatorhombated hecatonicosachoron is a uniform 4-polytope. It is composed of 2640 cells: 120 rhombicosidodecahedron, 600 truncated tetrahedra, 720 pentagonal prisms, and 1200 hexagonal prisms. It has 7200 vertices, 18000 edges, and 13440 faces (2400 triangles, 7200 squares, and 2400 hexagons).

Alternate names

Images

Schlegel diagram
Orthogonal projections in Coxeter planes

H3
A2/B3
A3/B2

Omnitruncated 120-cell

Omnitruncated 120-cell
TypeUniform 4-polytope
Uniform index46
Coxeter diagram
Cells2640 total:
120 4.6.10
720 4.4.10
1200 4.4.6
600 4.6.6
Faces17040 total:
10800 {4}, 4800 {6}
1440 {10}
Edges28800
Vertices14400
Vertex figure
Chiral scalene tetrahedron
Schläfli symbol t0,1,2,3{3,3,5}
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex

The omnitruncated 120-cell or great disprismatohexacosihecatonicosachoron is a convex uniform 4-polytope, composed of 2640 cells: 120 truncated icosidodecahedra, 600 truncated octahedra, 720 decagonal prisms, and 1200 hexagonal prisms. It has 14400 vertices, 28800 edges, and 17040 faces (10800 squares, 4800 hexagons, and 1440 decagons). It is the largest nonprismatic convex uniform 4-polytope.

The vertices and edges form the Cayley graph of the Coxeter group H4.

Alternate names

Images

Schlegel diagram (centered on truncated icosidodecahedron)
(Orthogonal view, centered on decagonal prism cell.)		 Stereographic projection
(centered on truncated icosidodecahedron)
Orthogonal projections in Coxeter planes

H3
A2/B3
A3/B2
Net

Omnitruncated 120-cell
Dual to omnitruncated 120-cell

Models

The first complete physical model of a 3D projection of the omnitruncated 120-cell was built by a team led by Daniel Duddy and David Richter on August 9, 2006 using the Zome system in the London Knowledge Lab for the 2006 Bridges Conference.[5]

Full snub 120-cell

Vertex figure for the omnisnub 120-cell

The full snub 120-cell or omnisnub 120-cell, defined as an alternation of the omnitruncated 120-cell, can not be made uniform, but it can be given Coxeter diagram , and symmetry [5,3,3]+, and constructed from 1200 octahedrons, 600 icosahedrons, 720 pentagonal antiprisms, 120 snub dodecahedrons, and 7200 tetrahedrons filling the gaps at the deleted vertices. It has 9840 cells, 35040 faces, 32400 edges, and 7200 vertices.[6]

Related polytopes

These polytopes are a part of a set of 15 uniform 4-polytopes with H4 symmetry:

Notes

  1. ↑ Klitizing, (x3o3o5x - sidpixhi)
  2. ↑ Klitizing, (x3o3x5x - prix)
  3. ↑ Klitizing, (x3x3o5x - prahi)
  4. ↑ Klitizing, (x3x3x5x - gidpixhi)
  5. ↑ Photos of Zome model of omnitruncated 120/600-cell
  6. ↑ http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm

References

External links

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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 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
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