Truncated 6-simplexes

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6-simplex t0.svg
6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t01.svg
Truncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t12.svg
Bitruncated 6-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t23.svg
Tritruncated 6-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Orthogonal projections in A7 Coxeter plane

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[edit]

Truncated 6-simplex
Type uniform 6-polytope
Class A6 polytope
Schläfli symbol t{3,3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel branch 11.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png
5-faces 14:
7 {3,3,3,3} 5-simplex t0.svg
7 t{3,3,3,3} 5-simplex t01.svg
4-faces 63:
42 {3,3,3} 4-simplex t0.svg
21 t{3,3,3} 4-simplex t01.svg
Cells 140:
105 {3,3} 3-simplex t0.svg
35 t{3,3} 3-simplex t01.svg
Faces 175:
140 {3}
35 {6}
Edges 126
Vertices 42
Vertex figure Elongated 5-cell pyramid
Coxeter group A6, [35], order 5040
Dual ?
Properties convex

Alternate names[edit]

  • Truncated heptapeton (Acronym: til) (Jonathan Bowers)[1]

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t01.svg 6-simplex t01 A5.svg 6-simplex t01 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t01 A3.svg 6-simplex t01 A2.svg
Dihedral symmetry [4] [3]

Bitruncated 6-simplex[edit]

Bitruncated 6-simplex
Type uniform 6-polytope
Class A6 polytope
Schläfli symbol 2t{3,3,3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png
5-faces 14
4-faces 84
Cells 245
Faces 385
Edges 315
Vertices 105
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names[edit]

  • Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)[2]

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t12.svg 6-simplex t12 A5.svg 6-simplex t12 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t12 A3.svg 6-simplex t12 A2.svg
Dihedral symmetry [4] [3]

Tritruncated 6-simplex[edit]

Tritruncated 6-simplex
Type uniform 6-polytope
Class A6 polytope
Schläfli symbol 3t{3,3,3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
or CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
5-faces 14 2t{3,3,3,3}
4-faces 84
Cells 280
Faces 490
Edges 420
Vertices 140
Vertex figure
Coxeter group A6, [[35]], order 10080
Properties convex, 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: CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png and CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png.

Alternate names[edit]

  • Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)[3]

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t23.svg 6-simplex t23 A5.svg 6-simplex t23 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph 6-simplex t23 A3.svg 6-simplex t23 A2.svg
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

Related polytopes[edit]

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
CDel branch 11.png = CDel node 1.pngCDel 6.pngCDel node.png
t{3} = {6}
Octahedron
CDel node 1.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
r{3,3} = {31,1} = {3,4}
Decachoron
CDel branch 11.pngCDel 3ab.pngCDel nodes.png
2t{33}
Dodecateron
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
2r{34} = {32,2}
Tetradecapeton
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
3t{35}
Hexadecaexon
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
3r{36} = {33,3}
Octadecazetton
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
4t{37}
Images Truncated triangle.png 3-cube t2.svgUniform polyhedron-33-t1.png 4-simplex t12.svgSchlegel half-solid bitruncated 5-cell.png 5-simplex t2.svg5-simplex t2 A4.svg 6-simplex t23.svg6-simplex t23 A5.svg 7-simplex t3.svg7-simplex t3 A5.svg 8-simplex t34.svg8-simplex t34 A7.svg
Facets {3} Regular polygon 3 annotated.svg t{3,3} Uniform polyhedron-33-t01.png r{3,3,3} Schlegel half-solid rectified 5-cell.png 2t{3,3,3,3} 5-simplex t12.svg 2r{3,3,3,3,3} 6-simplex t2.svg 3t{3,3,3,3,3,3} 7-simplex t23.svg
As
intersecting
dual
simplexes
Regular hexagon as intersection of two triangles.png
CDel branch 10.pngCDel branch 01.png
Stellated octahedron A4 A5 skew.png
CDel node.pngCDel split1.pngCDel nodes 10lu.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png
Compound dual 5-cells and bitruncated 5-cell intersection A4 coxeter plane.png
CDel branch.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes 01l.png
Dual 5-simplex intersection graph a5.pngDual 5-simplex intersection graph a4.png
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png

Related uniform 6-polytopes[edit]

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.

Notes[edit]

  1. ^ Klitzing, (o3x3o3o3o3o - til)
  2. ^ Klitzing, (o3x3x3o3o3o - batal)
  3. ^ Klitzing, (o3o3x3x3o3o - fe)

References[edit]

  • 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. "6D uniform polytopes (polypeta)".  o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds