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 |
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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 | |
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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.
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 | ||
---|---|---|
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.
Stereographic projection (Close up) |
H3 | A2 / B3 / D4 | A3 / B2 / D3 |
---|---|---|
[10] | [6] | [4] |
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.
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.
Centered on icosahedron | Centered on truncated tetrahedron | Central part and some of 120 red icosahedra. |
Net |
H4 | - | F4 |
---|---|---|
[30] | [20] | [12] |
H3 | A2 / B3 / D4 | A3 / B2 |
[10] | [6] | [4] |
H4 family polytopes | |||||||||||
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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} |
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.