Truncated 7-simplexes

7-simplex	Truncated 7-simplex
Bitruncated 7-simplex	Tritruncated 7-simplex
Orthogonal projections in A7 Coxeter plane

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

There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.

Truncated 7-simplex

Truncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces	16
5-faces
4-faces
Cells	350
Faces	336
Edges	196
Vertices	56
Vertex figure	( )v{3,3,3,3}
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex, Vertex-transitive

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

Alternate names

• Truncated octaexon (Acronym: toc) (Jonathan Bowers)

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

Bitruncated 7-simplex

Bitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	2t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	588
Vertices	168
Vertex figure	{ }v{3,3,3}
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex, Vertex-transitive

Alternate names

• Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)

Coordinates

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

Tritruncated 7-simplex

Tritruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	3t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	980
Vertices	280
Vertex figure	{3}v{3,3}
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex, Vertex-transitive

Alternate names

• Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)

Coordinates

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

Related polytopes

These three polytopes are from a set 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.  x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary