8-simplexTruncated 8-simplexRectified 8-simplexQuadritruncated 8-simplexTritruncated 8-simplexBitruncated 8-simplexOrthogonal projections in A8 Coxeter plane In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex.
There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex.
Truncated 8-simplex Truncated 8-simplexTypeuniform 8-polytopeSchläfli symbol t{37}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges288Vertices72Vertex figure( )v{3,3,3,3,3}Coxeter groupA8, , order 362880Propertiesconvex
Alternate names Truncated enneazetton (Acronym: tene) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the truncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex.
Images
Bitruncated 8-simplex Bitruncated 8-simplexTypeuniform 8-polytopeSchläfli symbol 2t{37}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges1008 Vertices252 Vertex figure{ }v{3,3,3,3}Coxeter groupA8, , order 362880Propertiesconvex
Alternate names Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the bitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 9-orthoplex.
Images
Tritruncated 8-simplex Tritruncated 8-simplexTypeuniform 8-polytopeSchläfli symbol 3t{37}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges2016Vertices504Vertex figure{3}v{3,3,3}Coxeter groupA8, , order 362880Propertiesconvex
Alternate names Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)Klitzing, (o3o3x3x3o3o3o3o - tatene)
Coordinates The Cartesian coordinates of the vertices of the tritruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 9-orthoplex.
Images
Quadritruncated 8-simplex Quadritruncated 8-simplexTypeuniform 8-polytopeSchläfli symbol 4t{37}Coxeter-Dynkin diagramsor 7-faces18 3t{3,3,3,3,3,3}6-faces5-faces4-facesCellsFacesEdges2520 Vertices630 Vertex figure{3,3}v{3,3}Coxeter groupA8, [], order 725760Propertiesconvex, isotopic The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets.
Alternate names Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the quadritruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,2,2,2). This construction is based on facets of the quadritruncated 9-orthoplex.
Images
Related polytopes
The four presented polytopes are in the family of 135 uniform 8-polytopes with A8 symmetry.
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. x3x3o3o3o3o3o3o - tene, o3x3x3o3o3o3o3o - batene, o3o3x3x3o3o3o3o - tatene, o3o3o3x3x3o3o3o - be
External links Polytopes of Various Dimensions Multi-dimensional Glossary