5-orthoplexRuncinated 5-orthoplexRuncinated 5-cubeRuncitruncated 5-orthoplexRuncicantellated 5-orthoplexRuncicantitruncated 5-orthoplexRuncitruncated 5-cubeRuncicantellated 5-cubeRuncicantitruncated 5-cubeOrthogonal projections in B5 Coxeter plane In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.
There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.
Runcinated 5-orthoplex Runcinated 5-orthoplexTypeUniform 5-polytopeSchläfli symbol t0,3{3,3,3,4}Coxeter-Dynkin diagram4-faces162Cells1200Faces2160Edges1440Vertices320Vertex figureCoxeter group B5 [4,3,3,3]D5 [32,1,1]Propertiesconvex
Alternate names Runcinated pentacross Small prismated triacontaditeron (Acronym: spat) (Jonathan Bowers)
Coordinates The vertices of the can be made in 5-space, as permutations and sign combinations of: (0,1,1,1,2)
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Runcitruncated 5-orthoplex Runcitruncated 5-orthoplexTypeuniform 5-polytopeSchläfli symbol t0,1,3{3,3,3,4}t0,1,3{3,31,1}Coxeter-Dynkin diagrams4-faces162Cells1440Faces3680Edges3360Vertices960Vertex figureCoxeter groupsB5, [3,3,3,4]D5, [32,1,1]Propertiesconvex
Alternate names Runcitruncated pentacross Prismatotruncated triacontaditeron (Acronym: pattit) (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of (±3,±2,±1,±1,0)
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Runcicantellated 5-orthoplex Runcicantellated 5-orthoplexTypeUniform 5-polytopeSchläfli symbol t0,2,3{3,3,3,4}t0,2,3{3,3,31,1}Coxeter-Dynkin diagram4-faces162Cells1200Faces2960Edges2880Vertices960Vertex figureCoxeter group B5 [4,3,3,3]D5 [32,1,1]Propertiesconvex
Alternate names Runcicantellated pentacross Prismatorhombated triacontaditeron (Acronym: pirt) (Jonathan Bowers)
Coordinates The vertices of the runcicantellated 5-orthoplex can be made in 5-space, as permutations and sign combinations of: (0,1,2,2,3)
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Runcicantitruncated 5-orthoplex Runcicantitruncated 5-orthoplexTypeUniform 5-polytopeSchläfli symbolt0,1,2,3{3,3,3,4}Coxeter-Dynkindiagram4-faces162Cells1440Faces4160Edges4800Vertices1920Vertex figureIrregular 5-cellCoxeter groups B5 [4,3,3,3]D5 [32,1,1]Propertiesconvex, isogonal
Alternate names Runcicantitruncated pentacross Great prismated triacontaditeron (gippit) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of a runcicantitruncated 5-orthoplex having an edge length of are given by all permutations of coordinates and sign of:
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Snub 5-demicube The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [32,1,1]+ or [4,(3,3,3)+], and constructed from 10 snub 24-cells, 32 snub 5-cells, 40 snub tetrahedral antiprisms, 80 2-3 duoantiprisms, and 960 irregular 5-cells filling the gaps at the deleted vertices.
Related polytopes These polytopes are from a set 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. x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit
External links
Polytopes of Various Dimensions, Jonathan Bowers Runcinated uniform polytera (spid), Jonathan Bowers Multi-dimensional Glossary