Orthogonal projections in A7 Coxeter plane
6-simplexTruncated 6-simplex
Bitruncated 6-simplexTritruncated 6-simplex

In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex.

Truncated 6-simplex

Truncated 6-simplex
Typeuniform 6-polytope
ClassA6 polytope
Schläfli symbolt{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces14: 7 {3,3,3,3} 7 t{3,3,3,3}
4-faces63: 42 {3,3,3} 21 t{3,3,3}
Cells140: 105 {3,3} 35 t{3,3}
Faces175: 140 {3} 35 {6}
Edges126
Vertices42
Vertex figure( )v{3,3,3}
Coxeter groupA6, [35], order 5040
Dual?
Propertiesconvex

Alternate names

  • Truncated heptapeton (Acronym: til) (Jonathan Bowers)

Coordinates

The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex.

Images

Orthographic projections
Ak Coxeter planeA6A5A4
Graph
Dihedral symmetry[7][6][5]
Ak Coxeter planeA3A2
Graph
Dihedral symmetry[4][3]

Bitruncated 6-simplex

Bitruncated 6-simplex
Typeuniform 6-polytope
ClassA6 polytope
Schläfli symbol2t{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces14
4-faces84
Cells245
Faces385
Edges315
Vertices105
Vertex figure{ }v{3,3}
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)

Coordinates

The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 7-orthoplex.

Images

Orthographic projections
Ak Coxeter planeA6A5A4
Graph
Dihedral symmetry[7][6][5]
Ak Coxeter planeA3A2
Graph
Dihedral symmetry[4][3]

Tritruncated 6-simplex

Tritruncated 6-simplex
Typeuniform 6-polytope
ClassA6 polytope
Schläfli symbol3t{3,3,3,3,3}
Coxeter-Dynkin diagramor
5-faces14 2t{3,3,3,3}
4-faces84
Cells280
Faces490
Edges420
Vertices140
Vertex figure{3}v{3}
Coxeter groupA6, [[35]], order 10080
Propertiesconvex, isotopic

The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets.

The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration: and .

Alternate names

  • Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)

Coordinates

The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,2). This construction is based on facets of the bitruncated 7-orthoplex. Alternately it can be centered on the origin as permutations of (-1,-1,-1,0,1,1,1).

Images

Orthographic projections
Ak Coxeter planeA6A5A4
Graph
Symmetry[[7]](*)=[14][6][[5]](*)=[10]
Ak Coxeter planeA3A2
Graph
Symmetry[4][[3]](*)=[6]

Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

Related polytopes

Isotopic uniform truncated simplices
Dim.2345678
Name CoxeterHexagon = t{3} = {6}Octahedron = r{3,3} = {31,1} = {3,4} { 3 3 } {\displaystyle \left\{{\begin{array}{l}3\\3\end{array}}\right\}}Decachoron 2t{33}Dodecateron 2r{34} = {32,2} { 3 , 3 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}Tetradecapeton 3t{35}Hexadecaexon 3r{36} = {33,3} { 3 , 3 , 3 3 , 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}}Octadecazetton 4t{37}
Images
Vertex figure( )∨( ){ }×{ }{ }∨{ }{3}×{3}{3}∨{3}{3,3}×{3,3}{3,3}∨{3,3}
Facets{3}t{3,3}r{3,3,3}2t{3,3,3,3}2r{3,3,3,3,3}3t{3,3,3,3,3,3}
As intersecting dual simplexes∩∩∩∩∩∩∩

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
t0t1t2t0,1t0,2t1,2t0,3t1,3t2,3
t0,4t1,4t0,5t0,1,2t0,1,3t0,2,3t1,2,3t0,1,4t0,2,4
t1,2,4t0,3,4t0,1,5t0,2,5t0,1,2,3t0,1,2,4t0,1,3,4t0,2,3,4t1,2,3,4
t0,1,2,5t0,1,3,5t0,2,3,5t0,1,4,5t0,1,2,3,4t0,1,2,3,5t0,1,2,4,5t0,1,2,3,4,5

Notes

  • 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 Ivić 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.
  • Klitzing, Richard. . o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe

External links

vteFundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations