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
Type	uniform 7-polytope
Schläfli symbol	rr{3,3,3,3,3,3} or ${\displaystyle r\left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}}$
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

• Small rhombated octaexon (acronym: saro) (Jonathan Bowers)^[1]

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.

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
Type	uniform 7-polytope
Schläfli symbol	r2r{3,3,3,3,3,3} or ${\displaystyle r\left\{{\begin{array}{l}3,3,3,3\\3,3\end{array}}\right\}}$
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

• Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)^[2]

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.

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
Type	uniform 7-polytope
Schläfli symbol	r3r{3,3,3,3,3,3} or ${\displaystyle r\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}}$
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

• Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)^[3]

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.

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
Type	uniform 7-polytope
Schläfli symbol	tr{3,3,3,3,3,3} or ${\displaystyle t\left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}}$
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

• Great rhombated octaexon (acronym: garo) (Jonathan Bowers)^[4]

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.

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
Type	uniform 7-polytope
Schläfli symbol	t2r{3,3,3,3,3,3} or ${\displaystyle t\left\{{\begin{array}{l}3,3,3,3\\3,3\end{array}}\right\}}$
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

• Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)^[5]

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.

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
Type	uniform 7-polytope
Schläfli symbol	t3r{3,3,3,3,3,3} or ${\displaystyle t\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}}$
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

• Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)^[6]

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.

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]]

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

A7 polytopes
t0	t1	t2	t3	t0,1	t0,2	t1,2	t0,3
t1,3	t2,3	t0,4	t1,4	t2,4	t0,5	t1,5	t0,6
t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4	t0,2,4	t1,2,4	t0,3,4
t1,3,4	t2,3,4	t0,1,5	t0,2,5	t1,2,5	t0,3,5	t1,3,5	t0,4,5
t0,1,6	t0,2,6	t0,3,6	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	t1,2,3,5	t0,1,4,5	t0,2,4,5	t1,2,4,5	t0,3,4,5
t0,1,2,6	t0,1,3,6	t0,2,3,6	t0,1,4,6	t0,2,4,6	t0,1,5,6	t0,1,2,3,4	t0,1,2,3,5
t0,1,2,4,5	t0,1,3,4,5	t0,2,3,4,5	t1,2,3,4,5	t0,1,2,3,6	t0,1,2,4,6	t0,1,3,4,6	t0,2,3,4,6
t0,1,2,5,6	t0,1,3,5,6	t0,1,2,3,4,5	t0,1,2,3,4,6	t0,1,2,3,5,6	t0,1,2,4,5,6	t0,1,2,3,4,5,6

• List of A7 polytopes

• 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, wiley.com, 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.
• Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh

• Polytopes of Various Dimensions
• Multi-dimensional Glossary