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.
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.
In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.
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.
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.
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.
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.
Ak Coxeter plane | A6 | A5 | A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
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 | 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 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
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 | 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:
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 |
---|---|---|---|
Graph | |||
Symmetry | [[7]](*)=[14] | [6] | [[5]](*)=[10] |
Ak Coxeter plane | A3 | A2 | |
Graph | |||
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 | |||||||
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 A6 Coxeter plane orthographic projections.
In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.
In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
In six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.
In six-dimensional geometry, a truncated 6-cube is a convex uniform 6-polytope, being a truncation of the regular 6-cube.
In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.
In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.
In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.
In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.
In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex.
In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.
In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.
In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex.
In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex.
In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.
In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.
In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.
In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.
In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
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Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 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 | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |