Truncated 120-cells

(Redirected from Truncated 600-cell)
 Orthogonal projections in H3 Coxeter plane 120-cell Truncated 120-cell Rectified 120-cell Bitruncated 120-cell Bitruncated 600-cell 600-cell Truncated 600-cell Rectified 600-cell

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

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 A3

[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

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

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.

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

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

Notes

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