7-simplexTruncated 7-simplexBitruncated 7-simplexTritruncated 7-simplexOrthogonal projections in A7 Coxeter plane In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.
Truncated 7-simplex Truncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces165-faces4-facesCells350Faces336Edges196Vertices56Vertex figure( )v{3,3,3,3}Coxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex, Vertex-transitive In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
Alternate names Truncated octaexon (acronym: toc) (Jonathan Bowers)
Coordinates The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex.
Images
Bitruncated 7-simplex Bitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol 2t{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges588 Vertices168 Vertex figure{ }v{3,3,3}Coxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex, Vertex-transitive
Alternate names Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)
Coordinates The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 8-orthoplex.
Images
Tritruncated 7-simplex Tritruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol 3t{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges980 Vertices280 Vertex figure{3}v{3,3}Coxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex, Vertex-transitive
Alternate names Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)
Coordinates The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 8-orthoplex.
Images
Related polytopes These three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry.
See also List of A7 polytopes
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. x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - bittoc, o3o3x3x3o3o3o - tattoc
External links Polytopes of Various Dimensions Multi-dimensional Glossary