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 $r\left\{\begin{array}{l}3, 3, 3, 3\\3\end{array}\right\}$
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, [3^5], order 5040
Properties	convex

Alternate names

• Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)^[1]

Coordinates

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]

^[2]

Bicantellated 6-simplex

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

Alternate names

• Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)^[3]

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 $t\left\{\begin{array}{l}3, 3, 3, 3\\3\end{array}\right\}$
Coxeter-Dynkin diagrams
5-faces	35
4-faces	210
Cells	560
Faces	805
Edges	630
Vertices	210
Vertex figure
Coxeter group	A6, [3^5], order 5040
Properties	convex

Alternate names

• Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)^[4]

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 $t\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\}$
Coxeter-Dynkin diagrams
5-faces	49
4-faces	329
Cells	980
Faces	1540
Edges	1260
Vertices	420
Vertex figure
Coxeter group	A6, [3^5], order 5040
Properties	convex

Alternate names

• Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)^[5]

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.

A6 polytopes 

t0	t1	t2	t0,1	t0,2	t1,2	t0,3	t1,3	t2,3
t0,4	t1,4	t0,5	t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4	t0,2,4
t1,2,4	t0,3,4	t0,1,5	t0,2,5	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,2,3,4	t1,2,3,4
t0,1,2,5	t0,1,3,5	t0,2,3,5	t0,1,4,5	t0,1,2,3,4	t0,1,2,3,5	t0,1,2,4,5	t0,1,2,3,4,5

Notes

1. Klitizing, (x3o3x3o3o3o - sril)
2. Klitzing, (x3o3x3o3o3o - sril)
3. Klitzing, (o3x3o3x3o3o - sabril)
4. Klitzing, (x3x3x3o3o3o - gril)
5. Klitzing, (o3x3x3x3o3o - gabril)

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
    • (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.
• Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril

External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary