5-cubeRectified 5-cubeBirectified 5-cubeBirectified 5-orthoplex5-orthoplexRectified 5-orthoplexOrthogonal projections in A5 Coxeter plane
In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex.
Rectified 5-orthoplex Rectified pentacrossTypeuniform 5-polytopeSchläfli symbol t1{3,3,3,4}Coxeter-Dynkin diagramsHypercells42 total:10 {3,3,4}32 t1{3,3,3}Cells240 total:80 {3,4}160 {3,3}Faces400 total:80+320 {3}Edges240Vertices40Vertex figureOctahedral prismPetrie polygonDecagonCoxeter groupsBC5, [3,3,3,4]D5, [32,1,1]Propertiesconvex
Its 40 vertices represent the root vectors of the simple Lie group D5. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B5 and C5 simple Lie groups.
E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr51 as a first rectification of a 5-dimensional cross polytope.
Alternate names Rectified pentacross Rectified triacontaditeron (32-faceted 5-polytope) Acronym: rat (Jonathan Bowers)
Construction There are two Coxeter groups associated with the rectified pentacross, one with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D5 or [32,1,1] Coxeter group.
Cartesian coordinates Cartesian coordinates for the vertices of a rectified pentacross, centered at the origin, edge length are all permutations of: (±1,±1,0,0,0)
Images
Related polytopes The rectified 5-orthoplex is the vertex figure for the 5-demicube honeycomb: or
This polytope is one 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. o3x3o3o4o - rat
External links Polytopes of Various Dimensions Multi-dimensional Glossary