Cantellated 6-simplexes

6-simplex	Cantellated 6-simplex	Bicantellated 6-simplex
Birectified 6-simplex	Cantitruncated 6-simplex	Bicantitruncated 6-simplex
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.

There are unique 4 degrees of cantellation for the 6-simplex, including truncations.

Cantellated 6-simplex

Cantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	rr{3,3,3,3,3} or [math]\displaystyle{ r\left\{\begin{array}{l}3, 3, 3, 3\\3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams
5-faces	35
4-faces	210
Cells	560
Faces	805
Edges	525
Vertices	105
Vertex figure	5-cell prism
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)

The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Bicantellated 6-simplex

Bicantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	2rr{3,3,3,3,3} or [math]\displaystyle{ r\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams
5-faces	49
4-faces	329
Cells	980
Faces	1540
Edges	1050
Vertices	210
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)

Coordinates

The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Cantitruncated 6-simplex

cantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	tr{3,3,3,3,3} or [math]\displaystyle{ t\left\{\begin{array}{l}3, 3, 3, 3\\3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams
5-faces	35
4-faces	210
Cells	560
Faces	805
Edges	630
Vertices	210
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)

Coordinates

The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Bicantitruncated 6-simplex

bicantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	2tr{3,3,3,3,3} or [math]\displaystyle{ t\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams
5-faces	49
4-faces	329
Cells	980
Faces	1540
Edges	1260
Vertices	420
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)

Coordinates

The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A₆ Coxeter plane orthographic projections.

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. "6D uniform polytopes (polypeta)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.  x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary