5-simplexRuncinated 5-simplexRuncitruncated 5-simplexBirectified 5-simplexRuncicantellated 5-simplexRuncicantitruncated 5-simplexOrthogonal projections in A5 Coxeter plane
In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.
There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.
Runcinated 5-simplex Runcinated 5-simplexTypeUniform 5-polytopeSchläfli symbol t0,3{3,3,3,3}Coxeter-Dynkin diagram4-faces476 t0,3{3,3,3} 20 {3}×{3}15 { }×r{3,3}6 r{3,3,3} Cells25545 {3,3} 180 { }×{3}30 r{3,3} Faces420240 {3} 180 {4} Edges270Vertices60Vertex figureCoxeter group A5 [3,3,3,3], order 720Propertiesconvex
Alternate names Runcinated hexateron Small prismated hexateron (Acronym: spix) (Jonathan Bowers)
Coordinates The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runcinated 6-orthoplex, or a biruncinated 6-cube respectively.
Images
Runcitruncated 5-simplex Runcitruncated 5-simplexTypeUniform 5-polytopeSchläfli symbolt0,1,3{3,3,3,3}Coxeter-Dynkin diagram4-faces476 t0,1,3{3,3,3}20 {3}×{6}15 { }×r{3,3}6 rr{3,3,3}Cells315Faces720Edges630Vertices180Vertex figureCoxeter groupA5 [3,3,3,3], order 720Propertiesconvex, isogonal
Alternate names Runcitruncated hexateron Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)
Coordinates The coordinates can be made in 6-space, as 180 permutations of: (0,0,1,1,2,3)
This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex.
Images
Runcicantellated 5-simplex Runcicantellated 5-simplexTypeUniform 5-polytopeSchläfli symbolt0,2,3{3,3,3,3}Coxeter-Dynkin diagram4-faces47Cells255Faces570Edges540Vertices180Vertex figureCoxeter groupA5 [3,3,3,3], order 720Propertiesconvex, isogonal
Alternate names Runcicantellated hexateron Biruncitruncated 5-simplex/hexateron Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)
Coordinates The coordinates can be made in 6-space, as 180 permutations of: (0,0,1,2,2,3)
This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex.
Images
Runcicantitruncated 5-simplex Runcicantitruncated 5-simplexTypeUniform 5-polytopeSchläfli symbolt0,1,2,3{3,3,3,3}Coxeter-Dynkin diagram4-faces476 t0,1,2,3{3,3,3}20 {3}×{6}15 {}×t{3,3} 6 tr{3,3,3}Cells31545 t0,1,2{3,3}120 { }×{3}120 { }×{6}30 t{3,3}Faces810120 {3}450 {4}240 {6}Edges900Vertices360Vertex figureIrregular 5-cellCoxeter groupA5 [3,3,3,3], order 720Propertiesconvex, isogonal
Alternate names Runcicantitruncated hexateron Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)
Coordinates The coordinates can be made in 6-space, as 360 permutations of: (0,0,1,2,3,4)
This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex.
Images
Related uniform 5-polytopes These polytopes are in a set of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
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. x3o3o3x3o - spix, x3x3o3x3o - pattix, x3o3x3x3o - pirx, x3x3x3x3o - gippix
External links
Polytopes of Various Dimensions, Jonathan Bowers Runcinated uniform polytera (spid), Jonathan Bowers Multi-dimensional Glossary