Cantellated 7-simplexes

7-simplex	Cantellated 7-simplex	Bicantellated 7-simplex	Tricantellated 7-simplex
Birectified 7-simplex	Cantitruncated 7-simplex	Bicantitruncated 7-simplex	Tricantitruncated 7-simplex
Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.

There are unique 6 degrees of cantellation for the 7-simplex, including truncations.

Cantellated 7-simplex

Cantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	rr{3,3,3,3,3,3} or [math]\displaystyle{ r\left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\} }[/math]
Coxeter-Dynkin diagram	or
6-faces
5-faces
4-faces
Cells
Faces
Edges	1008
Vertices	168
Vertex figure	5-simplex prism
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex

Alternate names

• Small rhombated octaexon (acronym: saro) (Jonathan Bowers)

The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Bicantellated 7-simplex

Bicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	r2r{3,3,3,3,3,3} or [math]\displaystyle{ r\left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams	or
6-faces
5-faces
4-faces
Cells
Faces
Edges	2520
Vertices	420
Vertex figure
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex

Alternate names

• Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)

Coordinates

The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Tricantellated 7-simplex

Tricantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	r3r{3,3,3,3,3,3} or [math]\displaystyle{ r\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams	or
6-faces
5-faces
4-faces
Cells
Faces
Edges	3360
Vertices	560
Vertex figure
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex

Alternate names

• Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)

Coordinates

The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Cantitruncated 7-simplex

Cantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	tr{3,3,3,3,3,3} or [math]\displaystyle{ t\left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	1176
Vertices	336
Vertex figure
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex

Alternate names

• Great rhombated octaexon (acronym: garo) (Jonathan Bowers)

Coordinates

The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Bicantitruncated 7-simplex

Bicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t2r{3,3,3,3,3,3} or [math]\displaystyle{ t\left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams	or
6-faces
5-faces
4-faces
Cells
Faces
Edges	2940
Vertices	840
Vertex figure
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex

Alternate names

• Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)

Coordinates

The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Tricantitruncated 7-simplex

Tricantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t3r{3,3,3,3,3,3} or [math]\displaystyle{ t\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams	or
6-faces
5-faces
4-faces
Cells
Faces
Edges	3920
Vertices	1120
Vertex figure
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex

Alternate names

• Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)

Coordinates

The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.

Related polytopes

This polytope is one of 71 uniform 7-polytopes with A₇ symmetry.

See also

• List of A7 polytopes

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. "7D uniform polytopes (polyexa)". https://bendwavy.org/klitzing/dimensions/polyexa.htm.  x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary