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Cantellated 5-simplexes
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In five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex.
There are unique 4 degrees of cantellation for the 5-simplex, including truncations.
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Cantellated 5-simplex
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Perspective
Cantellated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | rr{3,3,3,3} = | |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() or ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 27 | 6 r{3,3,3}![]() 6 rr{3,3,3} ![]() 15 {}x{3,3} ![]() |
Cells | 135 | 30 {3,3}![]() 30 r{3,3} ![]() 15 rr{3,3} ![]() 60 {}x{3} ![]() |
Faces | 290 | 200 {3} 90 {4} |
Edges | 240 | |
Vertices | 60 | |
Vertex figure | ![]() Tetrahedral prism | |
Coxeter group | A5 [3,3,3,3], order 720 | |
Properties | convex |
The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).
Alternate names
- Cantellated hexateron
- Small rhombated hexateron (Acronym: sarx) (Jonathan Bowers)[1]
Coordinates
The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,1,2) or of (0,1,1,2,2,2). These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively.
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Bicantellated 5-simplex
Summarize
Perspective
Bicantellated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | 2rr{3,3,3,3} = | |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() or ![]() ![]() ![]() ![]() ![]() | |
4-faces | 32 | 12 t02{3,3,3} 20 {3}x{3} |
Cells | 180 | 30 t1{3,3} 120 {}x{3} 30 t02{3,3} |
Faces | 420 | 240 {3} 180 {4} |
Edges | 360 | |
Vertices | 90 | |
Vertex figure | ![]() | |
Coxeter group | A5×2, [[3,3,3,3]], order 1440 | |
Properties | convex, isogonal |
Alternate names
- Bicantellated hexateron
- Small birhombated dodecateron (Acronym: sibrid) (Jonathan Bowers)[2]
Coordinates
The coordinates can be made in 6-space, as 90 permutations of:
- (0,0,1,1,2,2)
This construction exists as one of 64 orthant facets of the bicantellated 6-orthoplex.
Images
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Cantitruncated 5-simplex
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Perspective
cantitruncated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | tr{3,3,3,3} = | |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() or ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 27 | 6 t012{3,3,3}![]() 6 t{3,3,3} ![]() 15 {}x{3,3} |
Cells | 135 | 15 t012{3,3} ![]() 30 t{3,3} ![]() 60 {}x{3} 30 {3,3} ![]() |
Faces | 290 | 120 {3}![]() 80 {6} ![]() 90 {}x{} ![]() |
Edges | 300 | |
Vertices | 120 | |
Vertex figure | ![]() Irr. 5-cell | |
Coxeter group | A5 [3,3,3,3], order 720 | |
Properties | convex |
Alternate names
- Cantitruncated hexateron
- Great rhombated hexateron (Acronym: garx) (Jonathan Bowers)[3]
Coordinates
The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 6-cube respectively.
Images
Bicantitruncated 5-simplex
Summarize
Perspective
Bicantitruncated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | 2tr{3,3,3,3} = | |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() or ![]() ![]() ![]() ![]() ![]() | |
4-faces | 32 | 12 tr{3,3,3} 20 {3}x{3} |
Cells | 180 | 30 t{3,3} 120 {}x{3} 30 t{3,4} |
Faces | 420 | 240 {3} 180 {4} |
Edges | 450 | |
Vertices | 180 | |
Vertex figure | ![]() | |
Coxeter group | A5×2, [[3,3,3,3]], order 1440 | |
Properties | convex, isogonal |
Alternate names
- Bicantitruncated hexateron
- Great birhombated dodecateron(Acronym: gibrid) (Jonathan Bowers)[4]
Coordinates
The coordinates can be made in 6-space, as 180 permutations of:
- (0,0,1,2,3,3)
This construction exists as one of 64 orthant facets of the bicantitruncated 6-orthoplex.
Images
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Related uniform 5-polytopes
The cantellated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
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Notes
References
External links
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