Truncated 120-cells

120-cell File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:120-cell t01 H3.svg Truncated 120-cell File:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:120-cell t1 H3.svg Rectified 120-cell File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:120-cell t12 H3.png Bitruncated 120-cell Bitruncated 600-cell File:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png
File:600-cell t0 H3.svg 600-cell File:CDel node.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.png	File:600-cell t01 H3.svg Truncated 600-cell File:CDel node.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.png	File:600-cell t1 H3.svg Rectified 600-cell File:CDel node.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png
Orthogonal 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-cell
File:Schlegel half-solid truncated 120-cell.png Schlegel diagram (tetrahedron cells visible)
Type	Uniform 4-polytope
Uniform index	36
Schläfli symbol	t0,1{5,3,3} or t{5,3,3}
Coxeter diagrams	File:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Cells	600 3.3.3 File:Tetrahedron.png 120 3.10.10 File:Truncated dodecahedron.png
Faces	2400 triangles 720 decagons
Edges	4800
Vertices	2400
Vertex figure	File:Truncated 120-cell verf.png triangular pyramid
Dual	Tetrakis 600-cell
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

File:Truncated hecatonicosachoron net.png

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)

Images

Orthographic projections by Coxeter planes

H4	-	F4
File:120-cell t01 H4.svg [30]	File:120-cell t01 p20.svg [20]	File:120-cell t01 F4.svg [12]
H3	A2	A3
File:120-cell t01 H3.svg [10]	File:120-cell t01 A2.svg [6]	File:120-cell t01 A3.svg [4]

File:Truncated 120-cell net.png net

File:Truncated 120cell.png Central part of stereographic projection (centered on truncated dodecahedron)

File:Stereographic truncated 120-cell.png Stereographic projection

Bitruncated 120-cell

Bitruncated 120-cell
File:Bitruncated 120-cell schlegel halfsolid.png Schlegel diagram, centered on truncated icosahedron, truncated tetrahedral cells visible
Type	Uniform 4-polytope
Uniform index	39
Coxeter diagram	File:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png
Schläfli symbol	t1,2{5,3,3} or 2t{5,3,3}
Cells	720: 120 5.6.6 File:Truncated icosahedron.png 600 3.6.6 File:Truncated tetrahedron.png
Faces	4320: 1200{3}+720{5}+ 2400{6}
Edges	7200
Vertices	3600
Vertex figure	File:Bitruncated 120-cell verf.png digonal disphenoid
Symmetry group	H4, [3,3,5], order 14400
Properties	convex, vertex-transitive

File:Hexacosihecatonicosachoron net.png

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)

Images

File:Bitruncated cosmotetron stereographic close-up.png Stereographic projection (Close up)

Orthographic projections by Coxeter planes

H3	A2 / B3 / D4	A3 / B2 / D3
File:120-cell t12 H3.png [10]	File:120-cell t12 B3.png [6]	File:120-cell t12 A3.png [4]

Truncated 600-cell

Truncated 600-cell
File:Schlegel half-solid truncated 600-cell.png Schlegel diagram (icosahedral cells visible)
Type	Uniform 4-polytope
Uniform index	41
Schläfli symbol	t0,1{3,3,5} or t{3,3,5}
Coxeter diagram	File:CDel node.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.png
Cells	720: 120 File:Icosahedron.png 3.3.3.3.3 600 File:Truncated tetrahedron.png 3.6.6
Faces	2400{3}+1200{6}
Edges	4320
Vertices	1440
Vertex figure	File:Truncated 600-cell verf.png pentagonal pyramid
Dual	Dodecakis 120-cell
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

File:Truncated hexacosichoron net.png

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)
• 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 diagrams

File:Stereographic truncated 600-cell.png Centered on icosahedron

File:Truncated 600 cell.png Centered on truncated tetrahedron

File:Truncated 600 cell central.png Central part and some of 120 red icosahedra.

File:Truncated 600-cell net.png Net

Orthographic projections by Coxeter planes

H4	-	F4
File:600-cell t01 H4.svg [30]	File:600-cell t01 p20.svg [20]	File:600-cell t01 F4.svg [12]
H3	A2 / B3 / D4	A3 / B2
File:600-cell t01 H3.svg [10]	File:600-cell t01 A2.svg [6]	File:600-cell t01.svg [4]

3D Parallel projection
File:Truncated 600-cell parallel-icosahedron-first-01.png	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.

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 [1] m58 m59 m53
• Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 36, 39, 41, George Olshevsky.
• 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}