Runcinated 120-cells explained
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
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)[1]
Images
Runcitruncated 120-cell
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)[2]
Images
Runcitruncated 600-cell
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)[3]
Images
Omnitruncated 120-cell
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)[4]
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
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:
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,
- (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 http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf m55 m62 m60 m64
- x3o3o5x - sidpixhi, x3o3x5x - prix, x3x3o5x - prahi, x3x3x5x - gidpixhi
External links
Notes and References
- Klitizing, (x3o3o5x - sidpixhi)
- Klitizing, (x3o3x5x - prix)
- Klitizing, (x3x3o5x - prahi)
- Klitizing, (x3x3x5x - gidpixhi)
- http://homepages.wmich.edu/~drichter/bridgeszome2006.htm Photos of Zome model of omnitruncated 120/600-cell
- Web site: S3s3s5s.
