Truncated 120-cells
Uniform 4-polytope
In geometry , a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell .
There are three truncations, including a bitruncation , and a tritruncation, which creates the truncated 600-cell .
Truncated 120-cell
Net
The truncated 120-cell or truncated hecatonicosachoron is a uniform 4-polytope , constructed by a uniform truncation of the regular 120-cell 4-polytope .
It is made of 120 truncated dodecahedral and 600 tetrahedral cells. It has 3120 faces: 2400 being triangles and 720 being decagons . There are 4800 edges of two types: 3600 shared by three truncated dodecahedra and 1200 are shared by two truncated dodecahedra and one tetrahedron. Each vertex has 3 truncated dodecahedra and one tetrahedron around it. Its vertex figure is an equilateral triangular pyramid.
Alternate names
Truncated 120-cell (Norman W. Johnson )
Tuncated hecatonicosachoron / Truncated dodecacontachoron / Truncated polydodecahedron
Truncated-icosahedral hexacosihecatonicosachoron (Acronym thi) (George Olshevsky, and Jonathan Bowers)[ 1]
Images
Bitruncated 120-cell
Bitruncated 120-cell
Schlegel diagram , centered on truncated icosahedron, truncated tetrahedral cells visible
Type
Uniform 4-polytope
Uniform index
39
Coxeter diagram
Schläfli symbol
t1,2 {5,3,3} or 2t{5,3,3}
Cells
720: 120 5.6.6 600 3.6.6
Faces
4320: 1200{3}+720{5}+ 2400{6}
Edges
7200
Vertices
3600
Vertex figure
digonal disphenoid
Symmetry group
H4 , [3,3,5], order 14400
Properties
convex , vertex-transitive
Net
The bitruncated 120-cell or hexacosihecatonicosachoron is a uniform 4-polytope . It has 720 cells: 120 truncated icosahedra , and 600 truncated tetrahedra . Its vertex figure is a digonal disphenoid , with two truncated icosahedra and two truncated tetrahedra around it.
Alternate names
Bitruncated 120-cell / Bitruncated 600-cell (Norman W. Johnson )
Bitruncated hecatonicosachoron / Bitruncated hexacosichoron / Bitruncated polydodecahedron / Bitruncated polytetrahedron
Truncated-icosahedral hexacosihecatonicosachoron (Acronym Xhi) (George Olshevsky, and Jonathan Bowers)[ 2]
Images
Truncated 600-cell
Net
The truncated 600-cell or truncated hexacosichoron is a uniform 4-polytope . It is derived from the 600-cell by truncation . It has 720 cells: 120 icosahedra and 600 truncated tetrahedra . Its vertex figure is a pentagonal pyramid , with one icosahedron on the base, and 5 truncated tetrahedra around the sides.
Alternate names
Truncated 600-cell (Norman W. Johnson )
Truncated hexacosichoron (Acronym tex) (George Olshevsky, and Jonathan Bowers)[ 3]
Truncated tetraplex (Conway)
Structure
The truncated 600-cell consists of 600 truncated tetrahedra and 120 icosahedra . The truncated tetrahedral cells are joined to each other via their hexagonal faces, and to the icosahedral cells via their triangular faces. Each icosahedron is surrounded by 20 truncated tetrahedra.
Images
3D Parallel projection
Parallel projection into 3 dimensions, centered on an icosahedron. Nearest icosahedron to the 4D viewpoint rendered in red, remaining icosahedra in yellow. Truncated tetrahedra in transparent green.
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
^ Klitizing, (o3o3x5x - thi)
^ Klitizing, (o3x3x5o - xhi)
^ Klitizing, (x3x3o5o - tex)
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 [1] m58 m59 m53
Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 36, 39, 41 , George Olshevsky.
Klitzing, Richard. "4D uniform polytopes (polychora)" . o3o3x5x - thi, o3x3x5o - xhi, x3x3o5o - tex
Four-Dimensional Polytope Projection Barn Raisings (A Zometool construction of the truncated 120-cell ), George W. Hart
External links