120-cell | Cantellated 120-cell | Cantellated 600-cell |
600-cell | Cantitruncated 120-cell | Cantitruncated 600-cell |
Orthogonal projections in H3 Coxeter plane |
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In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 120-cell.
There are four degrees of cantellations of the 120-cell including with permutations truncations. Two are expressed relative to the dual 600-cell.
Cantellated 120-cell | |
---|---|
Type | Uniform 4-polytope |
Uniform index | 37 |
Coxeter diagram | |
Cells | 1920 total: 120 (3.4.5.4) 1200 (3.4.4) 600 (3.3.3.3) |
Faces | 4800{3}+3600{4}+720{5} |
Edges | 10800 |
Vertices | 3600 |
Vertex figure | wedge |
Schläfli symbol | t0,2{5,3,3} |
Symmetry group | H4, [3,3,5], order 14400 |
Properties | convex |
The cantellated 120-cell is a uniform 4-polytope. It is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra, 1200 triangular prisms, 600 octahedra. Its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex.
H3 | A2 / B3 / D4 | A3 / B2 |
---|---|---|
[10] | [6] | [4] |
Schlegel diagram. Pentagonal face are removed. |
Cantitruncated 120-cell | |
---|---|
Type | Uniform 4-polytope |
Uniform index | 42 |
Schläfli symbol | t0,1,2{5,3,3} |
Coxeter diagram | |
Cells | 1920 total: 120 (4.6.10) 1200 (3.4.4) 600 (3.6.6) |
Faces | 9120: 2400{3}+3600{4}+ 2400{6}+720{10} |
Edges | 14400 |
Vertices | 7200 |
Vertex figure | sphenoid |
Symmetry group | H4, [3,3,5], order 14400 |
Properties | convex |
The cantitruncated 120-cell polychoron is a uniform polychoron.
This 4-polytope is related to the regular 120-cell. The cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells.
The image shows the 4-polytope drawn as a Schlegel diagram, which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the decagonal faces are hidden, allowing us to see the elemented projected inside.
H3 | A2 / B3 / D4 | A3 / B2 |
---|---|---|
[10] | [6] | [4] |
Centered on truncated icosidodecahedron cell with decagonal faces hidden. |
Cantellated 600-cell | |
---|---|
Type | Uniform 4-polytope |
Uniform index | 40 |
Schläfli symbol | t0,2{3,3,5} |
Coxeter diagram | |
Cells | 1440 total: 120 3.5.3.5 600 3.4.3.4 720 4.4.5 |
Faces | 8640 total: (1200+2400){3} +3600{4}+1440{5} |
Edges | 10800 |
Vertices | 3600 |
Vertex figure | isosceles triangular prism |
Symmetry group | H4, [3,3,5], order 14400 |
Properties | convex |
The cantellated 600-cell is a uniform 4-polytope. It has 1440 cells: 120 icosidodecahedra, 600 cuboctahedra, and 720 pentagonal prisms. Its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms.
This 4-polytope has cells at 3 of 4 positions in the fundamental domain, extracted from the Coxeter diagram by removing one node at a time:
Node | Order | Coxeter diagram | Cell | Picture |
---|---|---|---|---|
0 | 600 | Cantellated tetrahedron (Cuboctahedron) | ||
1 | 1200 | None (Degenerate triangular prism) | ||
2 | 720 | Pentagonal prism | ||
3 | 120 | Rectified dodecahedron (Icosidodecahedron) |
There are 1440 pentagonal faces between the icosidodecahedra and pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms. There are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra.
There are two classes of edges: 3-4-4, 3-4-5: 3600 have two squares and a triangle around it, and 7200 have one triangle, one square, and one pentagon.
H4 | - |
---|---|
[30] | [20] |
F4 | H3 |
[12] | [10] |
A2 / B3 / D4 | A3 / B2 |
[6] | [4] |
Stereographic projection with its 3600 green triangular faces and its 3600 blue square faces. |
Cantitruncated 600-cell | |
---|---|
Type | Uniform 4-polytope |
Uniform index | 45 |
Coxeter diagram | |
Cells | 1440 total: 120 (5.6.6) 720 (4.4.5) 600 (4.6.6) |
Faces | 8640: 3600{4}+1440{5}+ 3600{6} |
Edges | 14400 |
Vertices | 7200 |
Vertex figure | sphenoid |
Schläfli symbol | t0,1,2{3,3,5} |
Symmetry group | H4, [3,3,5], order 14400 |
Properties | convex |
The cantitruncated 600-cell is a uniform 4-polytope. It is composed of 1440 cells: 120 truncated icosahedra, 720 pentagonal prisms and 600 truncated octahedra. It has 7200 vertices, 14400 edges, and 8640 faces (3600 squares, 1440 pentagons, and 3600 hexagons). It has an irregular tetrahedral vertex figure, filled by one truncated icosahedron, one pentagonal prism and two truncated octahedra.
H3 | A2 / B3 / D4 | A3 / B2 |
---|---|---|
[10] | [6] | [4] |
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} |
In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.
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 four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation of the regular tesseract.
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 tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
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 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 cantellated 24-cell is a convex uniform 4-polytope, being a cantellation of the regular 24-cell.
In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell.
In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation 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 five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex.
In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.
In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.
In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.
In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.
In four-dimensional Euclidean geometry, the cantitruncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a cantitruncation of the regular 24-cell honeycomb, containing truncated tesseract, cantitruncated 24-cell, and tetrahedral prism cells.