6-simplex |
Truncated 6-simplex | |
Bitruncated 6-simplex |
Tritruncated 6-simplex | |
Orthogonal projections in A_{7} 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 | |
---|---|
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 | A_{6}, [3^{5}], 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.
A_{k} Coxeter plane | A_{6} | A_{5} | A_{4} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
A_{k} Coxeter plane | A_{3} | A_{2} | |
Graph | |||
Dihedral symmetry | [4] | [3] |
Bitruncated 6-simplex | |
---|---|
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 | A_{6}, [3^{5}], 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.
A_{k} Coxeter plane | A_{6} | A_{5} | A_{4} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
A_{k} Coxeter plane | A_{3} | A_{2} | |
Graph | |||
Dihedral symmetry | [4] | [3] |
Tritruncated 6-simplex | |
---|---|
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 | A_{6}, [[3^{5}]], 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 .
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).
A_{k} Coxeter plane | A_{6} | A_{5} | A_{4} |
---|---|---|---|
Graph | |||
Symmetry | [[7]]^{(*)}=[14] | [6] | [[5]]^{(*)}=[10] |
A_{k} Coxeter plane | A_{3} | A_{2} | |
Graph | |||
Symmetry | [4] | [[3]]^{(*)}=[6] |
Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
Name Coxeter |
Hexagon = t{3} = {6} |
Octahedron = r{3,3} = {3^{1,1}} = {3,4} |
Decachoron 2t{3^{3}} |
Dodecateron 2r{3^{4}} = {3^{2,2}} |
Tetradecapeton 3t{3^{5}} |
Hexadecaexon 3r{3^{6}} = {3^{3,3}} |
Octadecazetton 4t{3^{7}} |
Images | |||||||
Vertex figure | ( )v( ) | { }×{ } |
{ }v{ } |
{3}×{3} |
{3}v{3} |
{3,3}x{3,3} | {3,3}v{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 |
∩ |
∩ |
∩ |
∩ |
∩ | ∩ | ∩ |
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 A_{6} Coxeter plane orthographic projections.