Runcinated 120-cell

Runcinated 120-cell
Four runcinations
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 t03 H3.png
Runcinated 120-cell
(Expanded 120-cell)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
120-cell t013 H3.png
Runcitruncated 120-cell
CDel node 1.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 t023 H3.png
Runcitruncated 600-cell
(Expanded 600-cell)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
120-cell t0123 H3.png
Omnitruncated 120-cell
CDel node 1.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 runcinated 120-cell (or runcinated 600-cell) is a convex uniform polychoron, 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 polychoron, the 120-cell or 600-cell.

Contents


Runcinated 120-cell

Runcinated 120-cell
Type Uniform polychoron
Uniform index 38
Coxeter-Dynkin diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 2640 total:
120 5.5.5 Dodecahedron.png
720 4.4.5 Pentagonal prism.png
1200 4.4.3 Triangular prism.png
600 3.3.3 Tetrahedron.png
Faces 7440:
2400{3}+3600{4}+
1440{5}
Edges 7200
Vertices 2400
Vertex figure Runcinated 120-cell verf.png
Equilateral-triangular antipodium
Schläfli symbol t0,3{5,3,3}
Symmetry group H4, [3,3,5]
Properties convex

The runcinated 120-cell is a uniform polychoron. 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

  • Runcinated 120-cell / Runcinated 600-cell (Norman W. Johnson)
    • Runcinated hecatonicosachoron / Runcinated hexacosichoron / Runcinated polydodecahedron / Runcinated polytetrahedron
  • Small diprismatohexacosihecatonicosachoron (acronym: sidpixhi) (George Olshevsky, Jonathan Bowers)[1]

Images

Runcinated 120-cell.png
Schlegel diagram (Only tetrahedral cells shown)
120-cell t03 H3.png
H3
120-cell t03 B3.png
A2/B3
Orthogonal projections in Coxeter planes

Runcitruncated 120-cell

Runcitruncated 120-cell
Type Uniform polychoron
Uniform index 43
Coxeter-Dynkin diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 2640 total:
120 (3.10.10) Truncated dodecahedron.png

720 (4.4.10) Decagonal prism.png
1200 (3.4.4) Triangular prism.png
600 (3.4.3.4) Cuboctahedron.png

Faces 13440:
4800{3}+7200{4}+
1440{10}
Edges 18000
Vertices 7200
Vertex figure Runcitruncated 120-cell verf.png
Irregular rectangular pyramid
Schläfli symbol t0,1,3{5,3,3}
Symmetry group H4, [3,3,5]
Properties convex

The runciruncated 120-cell is a uniform polychoron. 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

Runcitruncated 120-cell.png
Schlegel diagram (Only triangular prisms shown)
120-cell t013 H3.png
H3
120-cell t013 B3.png
A2/B3
Orthogonal projections in Coxeter planes

Runcitruncated 600-cell

Runcitruncated 600-cell
Type Uniform polychoron
Uniform index 44
Coxeter-Dynkin diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells 2640 total:
120 3.4.5.4 Small rhombicosidodecahedron.png
720 4.4.5 Pentagonal prism.png
1200 4.4.6 Hexagonal prism.png
600 3.6.6 Truncated tetrahedron.png
Faces 13440:
2400{3}+7200{4}+
1440{5}+2400{6}
Edges 18000
Vertices 7200
Vertex figure Runcitruncated 600-cell verf.png
Trapezoidal pyramid
Schläfli symbol t0,1,3{3,3,5}
Symmetry group H4, [3,3,5]
Properties convex

The runcitruncated 600-cell is a uniform polychoron. 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

Runcitruncated 600-cell.png
Schlegel diagram
120-cell t023 H3.png
H3
120-cell t023 B3.png
A2/B3
Orthogonal projections in Coxeter planes

Omnitruncated 120-cell

Omnitruncated 120-cell
Type Uniform polychoron
Uniform index 46
Coxeter-Dynkin diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells 2640 total:
120 4.6.10 Great rhombicosidodecahedron.png
720 4.4.10 Decagonal prism.png
1200 4.4.6 Hexagonal prism.png
600 4.6.6 Truncated octahedron.png
Faces 17040 total:
10800 {4}, 4800 {6}
1440 {10}
Edges 28800
Vertices 14400
Vertex figure Omnitruncated 120-cell verf.png
Chiral scalene tetrahedron
Schläfli symbol t0,1,2,3{3,3,5}
Symmetry group H4, [3,3,5]
Properties convex

The omnitruncated 120-cell is a convex uniform polychoron, 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 polychoron.

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

Alternate names

  • Omnitruncated 120-cell / Omnitruncated 600-cell (Norman W. Johnson)
  • Omnitruncated hecatonicosachoron / Omnitruncated hexacosichoron / Omnitruncated polydodecahedron / Omnitruncated polytetrahedron
  • Great diprismatohexacosihecatonicosachoron (Acronym gidpixhi) (George Olshevsky, Jonathan Bowers)[4]

Images

Omnitruncated 120-cell wireframe.png Stereographic omnitruncated 120-cell.png
Schlegel diagram (centered on truncated icosidodecahedron)
(Orthogonal view, centered on decagonal prism cell.)
Stereographic projection
(centered on truncated icosidodecahedron)
120-cell t0123 H3.png
H3
120-cell t0123 B3.png
A2/B3
Orthogonal projections in Coxeter planes
Omnitruncated 120-cell net.png
Polychoric net

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]

Related polytopes

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

H4 family polytopes by name, Coxeter-Dynkin diagram, and Schläfli symbol
120-cell rectified
120-cell
truncated
120-cell
cantellated
120-cell
runcinated
120-cell
bitruncated
120-cell
cantitruncated
120-cell
runcitruncated
120-cell
omnitruncated
120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
{5,3,3} t1{5,3,3} t0,1{5,3,3} t0,2{5,3,3} t0,3{5,3,3} t1,2{5,3,3} t0,1,2{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
120-cell t0 H3.svg 120-cell t1 H3.svg 120-cell t01 H3.svg 120-cell t02 H3.png 120-cell t03 H3.png 120-cell t12 H3.png 120-cell t012 H3.png 120-cell t013 H3.png 120-cell t0123 H3.png
600-cell t0 H3.svg 600-cell t1 H3.svg 600-cell t01 H3.svg 600-cell t02 H3.svg 120-cell t03 H3.png 120-cell t023 H3.png
600-cell rectified
600-cell
truncated
600-cell
cantellated
600-cell
runcinated
600-cell
bitruncated
600-cell
cantitruncated
600-cell
runcitruncated
600-cell
omnitruncated
600-cell
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
{3,3,5} t1{3,3,5} t0,1{3,3,5} t0,2{3,3,5} t0,3{3,3,5} t1,2{3,3,5} t0,1,2{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}

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

References


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