![]() 6-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Truncated 6-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
![]() Bitruncated 6-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Tritruncated 6-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
Orthogonal projections in A7 Coxeter plane |
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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 | |
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Type | uniform 6-polytope |
Class | A6 polytope |
Schläfli symbol | t{3,3,3,3,3} |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5-faces | 14: 7 {3,3,3,3} ![]() 7 t{3,3,3,3} ![]() |
4-faces | 63: 42 {3,3,3} ![]() 21 t{3,3,3} ![]() |
Cells | 140: 105 {3,3} ![]() 35 t{3,3} ![]() |
Faces | 175: 140 {3} 35 {6} |
Edges | 126 |
Vertices | 42 |
Vertex figure | ![]() ( )v{3,3,3} |
Coxeter group | A6, [35], order 5040 |
Dual | ? |
Properties | convex |
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.
Ak Coxeter plane | A6 | A5 | A4 |
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Graph | ![]() |
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Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [4] | [3] |
Bitruncated 6-simplex | |
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Type | uniform 6-polytope |
Class | A6 polytope |
Schläfli symbol | 2t{3,3,3,3,3} |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5-faces | 14 |
4-faces | 84 |
Cells | 245 |
Faces | 385 |
Edges | 315 |
Vertices | 105 |
Vertex figure | ![]() { }v{3,3} |
Coxeter group | A6, [35], order 5040 |
Properties | convex |
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.
Ak Coxeter plane | A6 | A5 | A4 |
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Graph | ![]() |
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Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [4] | [3] |
Tritruncated 6-simplex | |
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Type | uniform 6-polytope |
Class | A6 polytope |
Schläfli symbol | 3t{3,3,3,3,3} |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() or ![]() ![]() ![]() ![]() ![]() |
5-faces | 14 2t{3,3,3,3} |
4-faces | 84 |
Cells | 280 |
Faces | 490 |
Edges | 420 |
Vertices | 140 |
Vertex figure | ![]() {3}v{3} |
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: and
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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).
Ak Coxeter plane | A6 | A5 | A4 |
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Graph | ![]() |
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Symmetry | [[7]](*)=[14] | [6] | [[5]](*)=[10] |
Ak Coxeter plane | A3 | A2 | |
Graph | ![]() |
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Symmetry | [4] | [[3]](*)=[6] |
Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
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Name Coxeter |
Hexagon![]() ![]() ![]() ![]() t{3} = {6} |
Octahedron![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() r{3,3} = {31,1} = {3,4} |
Decachoron![]() ![]() ![]() 2t{33} |
Dodecateron![]() ![]() ![]() ![]() ![]() 2r{34} = {32,2} |
Tetradecapeton![]() ![]() ![]() ![]() ![]() 3t{35} |
Hexadecaexon![]() ![]() ![]() ![]() ![]() ![]() ![]() 3r{36} = {33,3} |
Octadecazetton![]() ![]() ![]() ![]() ![]() ![]() ![]() 4t{37} |
Images | ![]() |
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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 |
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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.