Cantellated 5-simplexes

5-simplex Template:CDD	Cantellated 5-simplex Template:CDD	Bicantellated 5-simplex Template:CDD
Birectified 5-simplex Template:CDD	Cantitruncated 5-simplex Template:CDD	Bicantitruncated 5-simplex Template:CDD
Orthogonal projections in A5 Coxeter plane

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. Template:Clear

Cantellated 5-simplex

Cantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	rr{3,3,3,3} = $r\left\{\begin{array}{c}3,3,3\\ 3\end{array}\right\}$
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
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)

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.

Bicantellated 5-simplex

Bicantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	2rr{3,3,3,3} = $r\left\{\begin{array}{c}3,3\\ 3,3\end{array}\right\}$
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
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 Template:Not a typo (Acronym: Template:Not a typo) (Jonathan Bowers)

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.

Cantitruncated 5-simplex

cantitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	tr{3,3,3,3} = $t\left\{\begin{array}{c}3,3,3\\ 3\end{array}\right\}$
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
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: Template:Not a typo) (Jonathan Bowers)

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.

Bicantitruncated 5-simplex

Bicantitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	2tr{3,3,3,3} = $t\left\{\begin{array}{c}3,3\\ 3,3\end{array}\right\}$
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
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 Template:Not a typo(Acronym: Template:Not a typo) (Jonathan Bowers)

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.

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)

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, Template:ISBN [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.
• Template:KlitzingPolytopes x3o3x3o3o - sarx, o3x3o3x3o - Template:Not a typo, x3x3x3o3o - Template:Not a typo, o3x3x3x3o - Template:Not a typo

External links

• Template:PolyCell
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary