# Cantellated 5-simplexes

(Redirected from Bicantellated 5-simplex)
 Orthogonal projections in A5 Coxeter plane 5-simplex Cantellated 5-simplex Bicantellated 5-simplex Birectified 5-simplex Cantitruncated 5-simplex Bicantitruncated 5-simplex

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.

## Cantellated 5-simplex

 Cantellated 5-simplex Type Uniform 5-polytope Schläfli symbol rr{3,3,3,3} = ${\displaystyle r\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}}$ 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.

### Images

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [3]

## Bicantellated 5-simplex

 Bicantellated 5-simplex Type Uniform 5-polytope Schläfli symbol 2rr{3,3,3,3} = ${\displaystyle r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$ 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

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [[3]]=[6]

## Cantitruncated 5-simplex

 cantitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol tr{3,3,3,3} = ${\displaystyle t\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}}$ 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

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [3]

## Bicantitruncated 5-simplex

 Bicantitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol 2tr{3,3,3,3} = ${\displaystyle t\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$ 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

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [[3]]=[6]

## 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)

## Notes

1. ^ Klitizing, (x3o3x3o3o - sarx)
2. ^ Klitizing, (o3x3o3x3o - sibrid)
3. ^ Klitizing, (x3x3x3o3o - garx)
4. ^ Klitizing, (o3x3x3x3o - gibrid)

## References

• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• 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 [1]
• (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]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "5D uniform polytopes (polytera)". x3o3x3o3o - sarx, o3x3o3x3o - sibrid, x3x3x3o3o - garx, o3x3x3x3o - gibrid