Truncated 120-cells

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120-cell t0 H3.svg
120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
120-cell t01 H3.svg
Truncated 120-cell
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
120-cell t1 H3.svg
Rectified 120-cell
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
120-cell t12 H3.png
Bitruncated 120-cell
Bitruncated 600-cell
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
600-cell t0 H3.svg
600-cell
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
600-cell t01 H3.svg
Truncated 600-cell
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
600-cell t1 H3.svg
Rectified 600-cell
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel 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 trunctions, including a bitruncation, and a tritruncation, which creates the truncated 600-cell.

Truncated 120-cell[edit]

Truncated 120-cell
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 CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cells 600 3.3.3 Tetrahedron.png
120 3.10.10 Truncated dodecahedron.png
Faces 2400 triangles
720 decagons
Edges 4800
Vertices 2400
Vertex figure Truncated 120-cell verf.png
triangular pyramid
Dual Tetrakis 600-cell
Symmetry group H4, [3,3,5], order 14400
Properties convex

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[edit]

  • Truncated 120-cell (Norman W. Johnson)
    • Tuncated hecatonicosachoron / Truncated dodecacontachoron / Truncated polydodecahedron
  • Truncated-icosahedral hexacosihecatonicosachoron (Acronym thi) (George Olshevsky, and Jonathan Bowers)[1]

Images[edit]

Orthographic projections by Coxeter planes
H4 - F4
120-cell t01 H4.svg
[30]
120-cell t01 p20.svg
[20]
120-cell t01 F4.svg
[12]
H3 A2 A3
120-cell t01 H3.svg
[10]
120-cell t01 A2.svg
[6]
120-cell t01 A3.svg
[4]
Truncated 120-cell net.png
net
Truncated 120cell.png
Central part of stereographic projection
(centered on truncated dodecahedron)
Stereographic truncated 120-cell.png
Stereographic projection

Bitruncated 120-cell[edit]

Bitruncated 120-cell
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 CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schläfli symbol t1,2{5,3,3}
or 2t{5,3,3}
Cells 720:
120 5.6.6 Truncated icosahedron.png
600 3.6.6 Truncated tetrahedron.png
Faces 4320:
1200{3}+720{5}+
2400{6}
Edges 7200
Vertices 3600
Vertex figure Bitruncated 120-cell verf.png
digonal disphenoid
Symmetry group H4, [3,3,5], order 14400
Properties convex, 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[edit]

  • 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[edit]

Bitruncated cosmotetron stereographic close-up.png
Stereographic projection (Close up)
Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2 / D3
120-cell t12 H3.png
[10]
120-cell t12 B3.png
[6]
120-cell t12 A3.png
[4]

Truncated 600-cell[edit]

Truncated 600-cell
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 CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells 720:
120 Icosahedron.png 3.3.3.3.3
600 Truncated tetrahedron.png 3.6.6
Faces 2400{3}+1200{6}
Edges 4320
Vertices 1440
Vertex figure Truncated 600-cell verf.png
pentagonal pyramid
Dual Dodecakis 120-cell
Symmetry group H4, [3,3,5], order 14400
Properties convex

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[edit]

Structure[edit]

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[edit]

Stereographic projection or Schlegel diagrams
Stereographic truncated 600-cell.png
Centered on icosahedron
Truncated 600 cell.png
Centered on truncated tetrahedron
Truncated 600 cell central.png
Central part
and some of 120 red icosahedra.
Truncated 600-cell net.png
Net
Orthographic projections by Coxeter planes
H4 - F4
600-cell t01 H4.svg
[30]
600-cell t01 p20.svg
[20]
600-cell t01 F4.svg
[12]
H3 A2 / B3 / D4 A3 / B2
600-cell t01 H3.svg
[10]
600-cell t01 A2.svg
[6]
600-cell t01.svg
[4]
3D Parallel projection
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.

Related polytopes[edit]

Notes[edit]

  1. ^ Klitizing, (o3o3x5x - thi)
  2. ^ Klitizing, (o3x3x5o - xhi)
  3. ^ Klitizing, (x3x3o5o - tex)

References[edit]

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds