6-cubeRuncinated 6-cubeBiruncinated 6-cubeRuncinated 6-orthoplex6-orthoplexRuncitruncated 6-cubeBiruncitruncated 6-cubeRuncicantellated 6-orthoplexRuncicantellated 6-cubeBiruncitruncated 6-orthoplexRuncitruncated 6-orthoplexRuncicanti-truncated 6-cubeBiruncicanti-truncated 6-cubeRuncicanti-truncated 6-orthoplexOrthogonal projections in B6 Coxeter plane
In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.
There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. 5 are expressed relative to the dual 6-orthoplex.
Runcinated 6-cube Runcinated 6-cubeTypeUniform 6-polytopeSchläfli symbolt0,3{4,3,3,3,3}Coxeter-Dynkin diagram5-faces4-facesCellsFacesEdges7680Vertices1280Vertex figureCoxeter groupB6 [4,3,3,3,3]Propertiesconvex
Alternate names Small prismated hexeract (spox) (Jonathan Bowers)
Images
Biruncinated 6-cube Biruncinated 6-cubeTypeUniform 6-polytopeSchläfli symbolt1,4{4,3,3,3,3}Coxeter-Dynkin diagram5-faces4-facesCellsFacesEdges11520Vertices1920Vertex figureCoxeter groupB6 [4,3,3,3,3]Propertiesconvex
Alternate names Small biprismated hexeractihexacontatetrapeton (sobpoxog) (Jonathan Bowers)
Images
Runcitruncated 6-cube Runcitruncated 6-cubeTypeUniform 6-polytopeSchläfli symbolt0,1,3{4,3,3,3,3}Coxeter-Dynkin diagram5-faces4-facesCellsFacesEdges17280Vertices3840Vertex figureCoxeter groupB6 [4,3,3,3,3]Propertiesconvex
Alternate names Prismatotruncated hexeract (potax) (Jonathan Bowers)
Images
Biruncitruncated 6-cube Biruncitruncated 6-cubeTypeUniform 6-polytopeSchläfli symbolt1,2,4{4,3,3,3,3}Coxeter-Dynkin diagram5-faces4-facesCellsFacesEdges23040Vertices5760Vertex figureCoxeter groupB6 [4,3,3,3,3]Propertiesconvex
Alternate names Biprismatotruncated hexeract (boprag) (Jonathan Bowers)
Images
Runcicantellated 6-cube Runcicantellated 6-cubeTypeUniform 6-polytopeSchläfli symbolt0,2,3{4,3,3,3,3}Coxeter-Dynkin diagram5-faces4-facesCellsFacesEdges13440Vertices3840Vertex figureCoxeter groupB6 [4,3,3,3,3]Propertiesconvex
Alternate names Prismatorhombated hexeract (prox) (Jonathan Bowers)
Images
Runcicantitruncated 6-cube Runcicantitruncated 6-cubeTypeUniform 6-polytopeSchläfli symbolt0,1,2,3{4,3,3,3,3}Coxeter-Dynkin diagram5-faces4-facesCellsFacesEdges23040Vertices7680Vertex figureCoxeter groupB6 [4,3,3,3,3]Propertiesconvex
Alternate names Great prismated hexeract (gippox) (Jonathan Bowers)Klitzing, (o3o3x3x3x4x - gippox)
Images
Biruncicantitruncated 6-cube Biruncicantitruncated 6-cubeTypeUniform 6-polytopeSchläfli symbolt1,2,3,4{4,3,3,3,3}Coxeter-Dynkin diagram5-faces4-facesCellsFacesEdges23040Vertices5760Vertex figureCoxeter groupB6 [4,3,3,3,3]Propertiesconvex
Alternate names Great biprismated hexeractihexacontatetrapeton (gobpoxog) (Jonathan Bowers)
Images
Related polytopes These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube and 6-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. o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - gobpoxog
External links
Polytopes of Various Dimensions Multi-dimensional Glossary