Truncated 7-simplex

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.

Contents 1 Truncated 7-simplex 1.1 Alternate names 1.2 Coordinates 1.3 Images 2 Bitruncated 7-simplex 2.1 Alternate names 2.2 Coordinates 2.3 Images 3 Tritruncated 7-simplex 3.1 Alternate names 3.2 Coordinates 3.3 Images 4 Related polytopes 5 See also 6 Notes 7 References 8 External links

Truncated 7-simplex

Truncated 7-simplex
Type	uniform polyexon
Schläfli symbol	t0,1{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	Elongated 5-simplex pyramid
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)^[1]

Coordinates

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 polyexon
Schläfli symbol	t1,2{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	588
Vertices	168
Vertex figure
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex, Vertex-transitive

Alternate names

• Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)^[2]

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 polyexon
Schläfli symbol	t2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	980
Vertices	280
Vertex figure
Coxeter groups	A7, [3,3,3,3,3,3]
Properties	convex, Vertex-transitive

Alternate names

• Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)^[3]

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.

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

See also

• List of A7 polytopes

Notes

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, editied 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.
• Richard Klitzing, 7D, uniform polytopes (polyexa) x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc

External links

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

Family	An	BCn	Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes