Runcinated 120-cells

Four runcinations

File:120-cell t0 H3.svg 120-cell Template:CDD	File:120-cell t03 H3.png Runcinated 120-cell (Expanded 120-cell) Template:CDD	File:120-cell t013 H3.png Runcitruncated 120-cell Template:CDD
File:600-cell t0 H3.svg 600-cell Template:CDD	File:120-cell t023 H3.png Runcitruncated 600-cell Template:CDD	File:120-cell t0123 H3.png Omnitruncated 120-cell Template:CDD
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	Template:CDD
Cells	2640 total: 120 5.5.5 File:Dodecahedron.png 720 4.4.5 File:Pentagonal prism.png 1200 4.4.3 File:Triangular prism.png 600 3.3.3 File:Tetrahedron.png
Faces	7440: 2400{3}+3600{4}+ 1440{5}
Edges	7200
Vertices	2400
Vertex figure	File:Runcinated 120-cell verf.png Equilateral-triangular antipodium
Schläfli symbol	t0,3{5,3,3}
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

File:Small disprismatohexacosihecatonicosachoron net.png

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

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

Images

Schlegel diagram (Only tetrahedral cells shown)

File:Runcinated 120-cell.png

Polyhedral rings

File:Runcinated 120-cell-5-fold-ring-cells.png Cells on 5-fold axis

File:Runcinated 120-cell-3-fold-ring-cells.png Cells on 3-fold axis

File:Runcinated 120-cell-2-fold-ring-cells.png Cells on 2-fold axis

Orthogonal projections in Coxeter planes

File:120-cell t03 H3.png H3

File:120-cell t03 B3.png A2/B3

File:120-cell t03 A3.png A3/B2

Runcitruncated 120-cell

Runcitruncated 120-cell
Type	Uniform 4-polytope
Uniform index	43
Coxeter diagram	Template:CDD
Cells	2640 total: 120 (3.10.10) File:Truncated dodecahedron.png 720 (4.4.10) File:Decagonal prism.png 1200 (3.4.4) File:Triangular prism.png 600 (3.4.3.4) File:Cuboctahedron.png
Faces	13440: 4800{3}+7200{4}+ 1440{10}
Edges	18000
Vertices	7200
Vertex figure	File:Runcitruncated 120-cell verf.png Irregular rectangular pyramid
Schläfli symbol	t0,1,3{5,3,3}
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

File:Prismatorhombated hexacosichoron net.png

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

• Runcicantellated 600-cell (Norman W. Johnson)
• Prismatorhombated hexacosichoron (Acronym: prix) (George Olshevsky, Jonathan Bowers)

Images

Schlegel diagram (Only triangular prisms shown)

File:Runcitruncated 120-cell.png

Orthogonal projections in Coxeter planes

File:120-cell t013 H3.png H3

File:120-cell t013 B3.png A2/B3

File:120-cell t013 A3.png A3/B2

Runcitruncated 600-cell

Runcitruncated 600-cell
Type	Uniform 4-polytope
Uniform index	44
Coxeter diagram	Template:CDD
Cells	2640 total: 120 3.4.5.4 File:Small rhombicosidodecahedron.png 720 4.4.5 File:Pentagonal prism.png 1200 4.4.6 File:Hexagonal prism.png 600 3.6.6 File:Truncated tetrahedron.png
Faces	13440: 2400{3}+7200{4}+ 1440{5}+2400{6}
Edges	18000
Vertices	7200
Vertex figure	File:Runcitruncated 600-cell verf.png Trapezoidal pyramid
Schläfli symbol	t0,1,3{3,3,5}
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

File:Prismatorhombated hecatonicosachoron net.png

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

• Runcicantellated 120-cell (Norman W. Johnson)
• Prismatorhombated hecatonicosachoron (Acronym: prahi) (George Olshevsky, Jonathan Bowers)

Images

Schlegel diagram

File:Runcitruncated 600-cell.png

Orthogonal projections in Coxeter planes

File:120-cell t023 H3.png H3

File:120-cell t023 B3.png A2/B3

File:120-cell t023 A3.png A3/B2

Omnitruncated 120-cell

Omnitruncated 120-cell
Type	Uniform 4-polytope
Uniform index	46
Coxeter diagram	Template:CDD
Cells	2640 total: 120 4.6.10 File:Great rhombicosidodecahedron.png 720 4.4.10 File:Decagonal prism.png 1200 4.4.6 File:Hexagonal prism.png 600 4.6.6 File:Truncated octahedron.png
Faces	17040 total: 10800 {4}, 4800 {6} 1440 {10}
Edges	28800
Vertices	14400
Vertex figure	File:Omnitruncated 120-cell verf.png 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.

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)

Images

File:Omnitruncated 120-cell wireframe.png	File:Stereographic omnitruncated 120-cell.png
Schlegel diagram (centered on truncated icosidodecahedron) (Orthogonal view, centered on decagonal prism cell.)	Stereographic projection (centered on truncated icosidodecahedron)

Orthogonal projections in Coxeter planes

File:120-cell t0123 H3.png H3

File:120-cell t0123 B3.png A2/B3

File:120-cell t0123 A3.png A3/B2

Polyhedral rings

File:Omnitruncated 120-cell-5-fold-ring-cells.png Cells on 5-fold axis

File:Omnitruncated 120-cell-3-fold-ring-cells.png Cells on 3-fold axis

File:Omnitruncated 120-cell-2-fold-ring-cells.png Cells on 2-fold axis

Net

File:Omnitruncated 120-cell net.png Omnitruncated 120-cell

File:Dual gidpixhi net.png 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.

Full snub 120-cell

File:Snub 120-cell verf.png

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 Template:CDD, 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.

Related polytopes

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

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, Template:Isbn
  • (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 [1] m55 m62 m60 m64
• Template:PolyCell
• Template:KlitzingPolytopes 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}