Runcinated 120-cells
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
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)[ 1]
Images
Polyhedral rings
Runcitruncated 120-cell
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)[ 2]
Images
Runcitruncated 600-cell
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)[ 3]
Images
Omnitruncated 120-cell
Omnitruncated 120-cell
Type
Uniform 4-polytope
Uniform index
46
Coxeter diagram
Cells
2640 total:4.6.10 4.4.10 4.4.6 4.6.6
Faces
17040 total:{4} , 4800 {6} {10}
Edges
28800
Vertices
14400
Vertex figure
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)[ 4]
Images
Polyhedral rings
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]
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 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]
These polytopes are a part of a set of 15 uniform 4-polytopes with H4 symmetry:
H4 family polytopes
120-cell
rectified
truncated
cantellated
runcinated
cantitruncated runcitruncated omnitruncated
{5,3,3}
r{5,3,3}
t{5,3,3}
rr{5,3,3}
t0,3
tr{5,3,3}
t0,1,3
t0,1,2,3
600-cell
rectified
truncated
cantellated
bitruncated
cantitruncated runcitruncated omnitruncated
{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
t0,1,2,3
Notes
References
Kaleidoscopes: Selected Writings of H.S.M. Coxeter 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, 1965N.W. Johnson : The Theory of Uniform Polytopes and Honeycombs , Ph.D. Dissertation, University of Toronto, 1966Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [1] 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)" .
External links
