120-cellTruncated 120-cellRectified 120-cellBitruncated 120-cellBitruncated 600-cell600-cellTruncated 600-cellRectified 600-cellOrthogonal projections in H3 Coxeter plane 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 Truncated 120-cellSchlegel diagram(tetrahedron cells visible)TypeUniform 4-polytopeUniform index36Schläfli symbolt0,1{5,3,3}or t{5,3,3}Coxeter diagramsCells600 3.3.3 120 3.10.10 Faces2400 triangles720 decagonsEdges4800Vertices2400Vertex figuretriangular pyramidDualTetrakis 600-cellSymmetry groupH4, [3,3,5], order 14400Propertiesconvex
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)Klitizing, (o3o3x5x - thi)
Images
+ Orthographic projections by Coxeter planesH4 -F4H3A2 A3
netCentral part of stereographic projection(centered on truncated dodecahedron)Stereographic projection
Bitruncated 120-cell Bitruncated 120-cellSchlegel diagram, centered on truncated icosahedron, truncated tetrahedral cells visibleTypeUniform 4-polytopeUniform index39Coxeter diagramSchläfli symbol t1,2{5,3,3}or 2t{5,3,3}Cells720:120 5.6.6 600 3.6.6 Faces4320:1200{3}+720{5}+2400{6}Edges7200Vertices3600Vertex figuredigonal disphenoidSymmetry groupH4, [3,3,5], order 14400Propertiesconvex, vertex-transitive
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)Klitizing, (o3x3x5o - xhi)
Images
Stereographic projection (Close up) + Orthographic projections by Coxeter planesH3A2 / B3 / D4A3 / B2 / D3
Truncated 600-cell Truncated 600-cellSchlegel diagram(icosahedral cells visible)TypeUniform 4-polytopeUniform index41Schläfli symbolt0,1{3,3,5}or t{3,3,5}Coxeter diagramCells720:120 3.3.3.3.3600 3.6.6Faces2400{3}+1200{6}Edges4320Vertices1440Vertex figurepentagonal pyramidDualDodecakis 120-cellSymmetry groupH4, [3,3,5], order 14400Propertiesconvex
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)Klitizing, (x3x3o5o - tex) 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 + Stereographic projection or Schlegel diagramsCentered on icosahedronCentered on truncated tetrahedronCentral partand some of 120 red icosahedra.
Net
+ Orthographic projections by Coxeter planesH4 -F4H3A2 / B3 / D4A3 / B2
3D Parallel projectionParallel 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.
Related polytopes
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, (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 m58 m59 m53
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 H4 uniform polytopes with coordinates: t{3,3,5} t{5,3,3} 2t{5,3,3}