5-simplexTruncated 5-simplexBitruncated 5-simplexOrthogonal projections in A5 Coxeter plane In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.
There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.
Truncated 5-simplex Truncated 5-simplexTypeUniform 5-polytopeSchläfli symbol t{3,3,3,3}Coxeter-Dynkin diagram4-faces126 {3,3,3}6 t{3,3,3}Cells4530 {3,3}15 t{3,3}Faces8060 {3}20 {6}Edges75Vertices30Vertex figure( )v{3,3}Coxeter group A5 [3,3,3,3], order 720Propertiesconvex The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).
Alternate names Truncated hexateron (Acronym: tix) (Jonathan Bowers)
Coordinates The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.
Images
Bitruncated 5-simplex Bitruncated 5-simplexTypeUniform 5-polytopeSchläfli symbol 2t{3,3,3,3}Coxeter-Dynkin diagram4-faces126 2t{3,3,3}6 t{3,3,3}Cells6045 {3,3}15 t{3,3}Faces14080 {3}60 {6}Edges150Vertices60Vertex figure{ }v{3}Coxeter group A5 [3,3,3,3], order 720Propertiesconvex
Alternate names Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)
Coordinates The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.
Images
Related uniform 5-polytopes The truncated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple, magenta having progressively more vertices.)
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. x3x3o3o3o - tix, o3x3x3o3o - bittix
External links
Polytopes of Various Dimensions, Jonathan Bowers Truncated uniform polytera (tix), Jonathan Bowers Multi-dimensional Glossary