6-simplexTruncated 6-simplexBitruncated 6-simplexTritruncated 6-simplexOrthogonal 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 Truncated 6-simplexTypeuniform 6-polytopeClassA6 polytopeSchläfli symbol t{3,3,3,3,3}Coxeter-Dynkin diagram5-faces14:7 {3,3,3,3} 7 t{3,3,3,3} 4-faces63:42 {3,3,3} 21 t{3,3,3} Cells140:105 {3,3} 35 t{3,3} Faces175:140 {3}35 {6}Edges126Vertices42Vertex figure( )v{3,3,3}Coxeter groupA6, , order 5040Dual?Propertiesconvex
Alternate names Truncated heptapeton (Acronym: til) (Jonathan Bowers)
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.
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Bitruncated 6-simplex Bitruncated 6-simplexTypeuniform 6-polytopeClassA6 polytopeSchläfli symbol 2t{3,3,3,3,3}Coxeter-Dynkin diagram5-faces144-faces84Cells245Faces385Edges315Vertices105Vertex figure{ }v{3,3}Coxeter groupA6, , order 5040Propertiesconvex
Alternate names Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)
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.
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Tritruncated 6-simplex Tritruncated 6-simplexTypeuniform 6-polytopeClassA6 polytopeSchläfli symbol 3t{3,3,3,3,3}Coxeter-Dynkin diagramor 5-faces14 2t{3,3,3,3}4-faces84 Cells280 Faces490 Edges420 Vertices140 Vertex figure{3}v{3}Coxeter groupA6, [], order 10080Propertiesconvex, 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 .
Alternate names Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)
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).
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Related 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.
Notes
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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, (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. o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe
External links Polytopes of Various Dimensions Multi-dimensional Glossary