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Truncated 120-cells

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120-cell
Truncated 120-cell
Rectified 120-cell
Bitruncated 120-cell
Bitruncated 600-cell

600-cell
Truncated 600-cell
Rectified 600-cell
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 trunctions, including a bitruncation, and a tritruncation, which creates the truncated 600-cell.

Truncated 120-cell

Truncated 120-cell

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
Cells 600 3.3.3
120 3.10.10
Faces 2400 triangles
720 decagons
Edges 4800
Vertices 2400
Vertex figure
triangular pyramid
Dual Tetrakis 600-cell
Symmetry group H4, [3,3,5], order 14400
Properties convex
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

Orthographic projections by Coxeter planes
H4 - F4

[30]
[20]
[12]
H3 A2 / B3 / D4 A3 / B2

[10]
[6]
[4]

net
Central part of stereographic projection
(centered on truncated dodecahedron)
Stereographic projection

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


Stereographic projection (Close up)
Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2 / D3

[10]
[6]
[4]

Truncated 600-cell

Truncated 600-cell

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
Cells 720:
120 3.3.3.3.3
600 3.6.6
Faces 2400{3}+1200{6}
Edges 4320
Vertices 1440
Vertex figure
pentagonal pyramid
Dual Dodecakis 120-cell
Symmetry group H4, [3,3,5], order 14400
Properties convex
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

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

Centered on icosahedron
Centered on truncated tetrahedron
Central part
and some of 120 red icosahedra.

Net
Orthographic projections by Coxeter planes
H4 - F4

[30]
[20]
[12]
H3 A2 / B3 / D4 A3 / B2

[10]
[6]
[4]
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

  1. ^ Klitizing, (o3o3x5x - thi)
  2. ^ Klitizing, (o3x3x5o - xhi)
  3. ^ 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
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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