5-orthoplexTruncated 5-orthoplexBitruncated 5-orthoplex5-cubeTruncated 5-cubeBitruncated 5-cubeOrthogonal projections in B5 Coxeter plane
In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.
There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.
Truncated 5-orthoplex Truncated 5-orthoplexTypeuniform 5-polytopeSchläfli symbolt{3,3,3,4}t{3,31,1}Coxeter-Dynkin diagrams4-faces4210 32 Cells240160 80 Faces400320 80 Edges280240 40 Vertices80Vertex figure( )v{3,4}Coxeter groupsB5, [3,3,3,4], order 3840D5, [32,1,1], order 1920Propertiesconvex
Alternate names Truncated pentacross Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of (±2,±1,0,0,0)
Images The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.
Bitruncated 5-orthoplex Bitruncated 5-orthoplexTypeuniform 5-polytopeSchläfli symbol2t{3,3,3,4}2t{3,31,1}Coxeter-Dynkin diagrams4-faces4210 32 Cells28040 160 80 Faces720320 320 80 Edges720480 240 Vertices240Vertex figure{ }v{4}Coxeter groupsB5, [3,3,3,4], order 3840D5, [32,1,1], order 1920Propertiesconvex The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb.
Alternate names Bitruncated pentacross Bitruncated triacontaditeron (Acronym: bittit) (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of (±2,±2,±1,0,0)
Images The bitruncated 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex.
Related polytopes The truncated 5-orthoplex and bitruncated 5-orthoplex are from the family of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
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. x3x3o3o4o - tot, o3x3x3o4o - bittit
External links
Polytopes of Various Dimensions Multi-dimensional Glossary