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
Type	Uniform 4-polytope
Uniform index	38
Coxeter 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], order 14400
Properties	convex

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.

Images

Schlegel diagram (Only tetrahedral cells shown)

Polyhedral rings

Cells on 5-fold axis

Cells on 3-fold axis

Cells on 2-fold axis

Orthogonal projections in Coxeter planes

H3

A2/B3

A3/B2

Runcitruncated 120-cell

Runcitruncated 120-cell
Type	Uniform 4-polytope
Uniform index	43
Coxeter 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], order 14400
Properties	convex

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.

Images

Schlegel diagram (Only triangular prisms shown)

Orthogonal projections in Coxeter planes

H3

A2/B3

A3/B2

Runcitruncated 600-cell

Runcitruncated 600-cell
Type	Uniform 4-polytope
Uniform index	44
Coxeter 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], order 14400
Properties	convex

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).

Images

Schlegel diagram

Orthogonal projections in Coxeter planes

H3

A2/B3

A3/B2

Omnitruncated 120-cell

Omnitruncated 120-cell
Type	Uniform 4-polytope
Uniform index	46
Coxeter 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], order 14400
Properties	convex

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.

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

Polyhedral rings

Cells on 5-fold axis

Cells on 3-fold axis

Cells on 2-fold axis

Net

Omnitruncated 120-cell

Dual to omnitruncated 120-cell

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:

H4 family polytopes
120-cell	rectified 120-cell	truncated 120-cell	cantellated 120-cell	runcinated 120-cell	cantitruncated 120-cell	runcitruncated 120-cell	omnitruncated 120-cell
{5,3,3}	r{5,3,3}	t{5,3,3}	rr{5,3,3}	t0,3{5,3,3}	tr{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	bitruncated 600-cell	cantitruncated 600-cell	runcitruncated 600-cell	omnitruncated 600-cell
{3,3,5}	r{3,3,5}	t{3,3,5}	rr{3,3,5}	2t{3,3,5}	tr{3,3,5}	t0,1,3{3,3,5}	t0,1,2,3{3,3,5}

Notes

References

• 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
  • (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• 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 m55 m62 m60 m64
• Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 38, 44, 47, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora)". x3o3o5x - sidpixhi, x3o3x5x - prix, x3x3o5x - prahi, x3x3x5x - gidpixhi

External links

• H4 uniform polytopes with coordinates: t03{5,3,3} t013{3,3,5} t013{5,3,3} t0123{5,3,3}