5-cubeRuncinated 5-cubeRuncinated 5-orthoplexRuncitruncated 5-cubeRuncicantellated 5-cubeRuncicantitruncated 5-cubeRuncitruncated 5-orthoplexRuncicantellated 5-orthoplexRuncicantitruncated 5-orthoplexOrthogonal projections in B5 Coxeter plane In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.
There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.
Runcinated 5-cube Runcinated 5-cubeTypeUniform 5-polytopeSchläfli symbol t0,3{4,3,3,3}Coxeter diagram4-faces20210 80 80 32 Cells124040 240 320 160 320 160 Faces2160240 960 640 320 Edges1440480+960Vertices320Vertex figureCoxeter group B5 [4,3,3,3]Propertiesconvex
Alternate names Small prismated penteract (Acronym: span) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:
Images
Runcitruncated 5-cube Runcitruncated 5-cubeTypeUniform 5-polytopeSchläfli symbolt0,1,3{4,3,3,3}Coxeter-Dynkin diagrams4-faces20210 80 80 32 Cells156040 240 320 320 160 320 160 Faces3760240 960 320 960 640 640 Edges3360480+960+1920Vertices960Vertex figureCoxeter groupB5, [3,3,3,4]Propertiesconvex
Alternate names Runcitruncated penteract Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)
Construction and coordinates The Cartesian coordinates of the vertices of a runcitruncated 5-cube having edge length 2 are all permutations of:
Images
Runcicantellated 5-cube Runcicantellated 5-cubeTypeUniform 5-polytopeSchläfli symbol t0,2,3{4,3,3,3}Coxeter-Dynkin diagram4-faces20210 80 80 32 Cells124040 240 320 320 160 160 Faces2960240 480 960 320 640 320 Edges2880960+960+960Vertices960Vertex figureCoxeter group B5 [4,3,3,3]Propertiesconvex
Alternate names Runcicantellated penteract Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:
Images
Runcicantitruncated 5-cube Runcicantitruncated 5-cubeTypeUniform 5-polytopeSchläfli symbolt0,1,2,3{4,3,3,3}Coxeter-Dynkindiagram4-faces202Cells1560Faces4240Edges4800Vertices1920Vertex figureIrregular 5-cellCoxeter group B5 [4,3,3,3]Propertiesconvex, isogonal
Alternate names Runcicantitruncated penteract Biruncicantitruncated pentacross Great prismated penteract (Acronym: ) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Related polytopes These polytopes are a part of a set of 31 uniform polytera 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. o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin
External links
Polytopes of Various Dimensions, Jonathan Bowers Runcinated uniform polytera (spid), Jonathan Bowers Multi-dimensional Glossary