Runcinated 120-cell

Four runcinations

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

600-cell
Runcitruncated 600-cell
(Expanded 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 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
Cells 2640 total:
120 5.5.5
720 4.4.5
1200 4.4.3
600 3.3.3
Faces 7440:
2400{3}+3600{4}+
1440{5}
Edges 7200
Vertices 2400
Vertex figure
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

Images

Schlegel diagram (Only tetrahedral cells shown)

H3
A2/B3
Orthogonal projections in Coxeter planes

Runcitruncated 120-cell

Runcitruncated 120-cell
Type Uniform polychoron
Uniform index 43
Coxeter-Dynkin diagram
Cells 2640 total:
120 (3.10.10)

720 (4.4.10)
1200 (3.4.4)
600 (3.4.3.4)
Faces 13440:
4800{3}+7200{4}+
1440{10}
Edges 18000
Vertices 7200
Vertex figure
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

Schlegel diagram (Only triangular prisms shown)

H3
A2/B3
Orthogonal projections in Coxeter planes

Runcitruncated 600-cell

Runcitruncated 600-cell
Type Uniform polychoron
Uniform index 44
Coxeter-Dynkin diagram
Cells 2640 total:
120 3.4.5.4
720 4.4.5
1200 4.4.6
600 3.6.6
Faces 13440:
2400{3}+7200{4}+
1440{5}+2400{6}
Edges 18000
Vertices 7200
Vertex figure
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

Schlegel diagram

H3
A2/B3
Orthogonal projections in Coxeter planes

Omnitruncated 120-cell

Omnitruncated 120-cell
Type Uniform polychoron
Uniform index 46
Coxeter-Dynkin diagram
Cells 2640 total:
120 4.6.10
720 4.4.10
1200 4.4.6
600 4.6.6
Faces 17040 total:
10800 {4}, 4800 {6}
1440 {10}
Edges 28800
Vertices 14400
Vertex figure
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

Images

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

H3
A2/B3
Orthogonal projections in Coxeter planes

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
{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}
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
{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