| 释义 |
- Truncated 6-simplex Alternate names Coordinates Images
- Bitruncated 6-simplex Alternate names Coordinates Images
- Tritruncated 6-simplex Alternate names Coordinates Images Related polytopes
- Related uniform 6-polytopes
- Notes
- References
- External links
6-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node}} | Truncated 6-simplex
{{CDD>node_1|3|node_1|3|node|3|node|3|node|3|node}} | Bitruncated 6-simplex
{{CDD>node|3|node_1|3|node_1|3|node|3|node|3|node}} | Tritruncated 6-simplex
{{CDD>node|3|node|3|node_1|3|node_1|3|node|3|node}} | | 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 | Truncated 6-simplex | | Type | uniform 6-polytope | | Class | A6 polytope | | Schläfli symbol | t{3,3,3,3,3} | | Coxeter-Dynkin diagram | node_1|3|node_1|3|node|3|node|3|node|3|node}}
{{CDD|branch_11|3b|nodeb|3b|nodeb|3b|nodeb|3b|nodeb}} | | 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 |
Alternate names - Truncated heptapeton (Acronym: til) (Jonathan Bowers)[1]
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 {{6-simplex Coxeter plane graphs|t01|150}}{{-}} Bitruncated 6-simplex | Bitruncated 6-simplex | | Type | uniform 6-polytope | | Class | A6 polytope | | Schläfli symbol | 2t{3,3,3,3,3} | | Coxeter-Dynkin diagram | node|3|node_1|3|node_1|3|node|3|node|3|node}}
{{CDD|branch_11|3ab|nodes|3b|nodeb|3b|nodeb}} | | 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 |
Alternate names - Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)[2]
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 {{6-simplex Coxeter plane graphs|t12|150}} Tritruncated 6-simplex | Tritruncated 6-simplex | | Type | uniform 6-polytope | | Class | A6 polytope | | Schläfli symbol | 3t{3,3,3,3,3} | | Coxeter-Dynkin diagram | {{CDD | 3|node | node_1|3|node_1|3|node|3|node}}
or {{CDD|branch_11|3ab|nodes|3ab|nodes}} | | 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: {{CDD|branch|3ab|nodes|3ab|nodes_10l}} and {{CDD|branch|3ab|nodes|3ab|nodes_01l}}. Alternate names - Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)[3]
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 {{6-simplex2 Coxeter plane graphs|t23|150}} Related polytopes{{Isotopic uniform simplex polytopes}} 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. {{Heptapeton family}} Notes1. ^Klitzing, (o3x3o3o3o3o - til)
2. ^Klitzing, (o3x3x3o3o3o - batal)
3. ^Klitzing, (o3o3x3x3o3o - fe)
References- 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}}
- (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.
- {{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}} o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe
External links - [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
- Multi-dimensional Glossary
{{Polytopes}} 1 : 6-polytopes |