Truncated 120-cells

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 Orthogonal projections in H3 Coxeter plane 120-cell Truncated 120-cell Rectified 120-cell Bitruncated 120-cellBitruncated 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.

Contents

There are three truncations, 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 index36
Schläfli symbol t0,1{5,3,3}
or t{5,3,3}
Coxeter diagrams
Cells600 3.3.3
120 3.10.10
Faces2400 triangles
720 decagons
Edges4800
Vertices2400
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]
H3A2A3

[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 index39
Coxeter diagram
Schläfli 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}
Edges7200
Vertices3600
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
H3A2 / B3 / D4A3 / B2 / D3

[10]

[6]

[4]

Truncated 600-cell

Truncated 600-cell

Schlegel diagram
(icosahedral cells visible)
Type Uniform 4-polytope
Uniform index41
Schläfli symbol t0,1{3,3,5}
or t{3,3,5}
Coxeter diagram
Cells720:
120 3.3.3.3.3
600 3.6.6
Faces2400{3}+1200{6}
Edges4320
Vertices1440
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.

Alternate names

• Truncated 600-cell (Norman W. Johnson)
• Truncated hexacosichoron (Acronym tex) (George Olshevsky, and Jonathan Bowers) [3]
• 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

 Centered on icosahedron Centered on truncated tetrahedron Central partand some of 120 red icosahedra.
 Net
Orthographic projections by Coxeter planes
H4-F4

[30]

[20]

[12]
H3A2 / B3 / D4A3 / 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)

Related Research Articles

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.

In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.

In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.

In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In geometry, a rectified 120-cell is a uniform 4-polytope formed as the rectification of the regular 120-cell.

In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation of the regular 5-cell.

In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation of the regular 120-cell.

In four-dimensional geometry, a runcinated 120-cell is a convex uniform 4-polytope, being a runcination of the regular 120-cell.

In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

In 4-dimensional geometry, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 cells It has 64 faces, and 96 edges and 48 vertices.

In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

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 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
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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