+ Four rectifications120-cellRectified 120-cell600-cellRectified 600-cellOrthogonal projections in H3 Coxeter plane In geometry, a rectified 120-cell is a uniform 4-polytope formed as the rectification of the regular 120-cell.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC120.
There are four rectifications of the 120-cell, including the zeroth, the 120-cell itself. The birectified 120-cell is more easily seen as a rectified 600-cell, and the trirectified 120-cell is the same as the dual 600-cell.
Rectified 120-cell Rectified 120-cellSchlegel diagram, centered on icosidodecahedon, tetrahedral cells visibleTypeUniform 4-polytopeUniform index33Coxeter diagramSchläfli symbol t1{5,3,3}or r{5,3,3}Cells720 total:120 (3.5.3.5) 600 (3.3.3) Faces3120 total:2400 {3}, 720 {5}Edges3600Vertices1200Vertex figuretriangular prismSymmetry groupH4 or [3,3,5]Propertiesconvex, vertex-transitive, edge-transitive
In geometry, the rectified 120-cell or rectified hecatonicosachoron is a convex uniform 4-polytope composed of 600 regular tetrahedra and 120 icosidodecahedra cells. Its vertex figure is a triangular prism, with three icosidodecahedra and two tetrahedra meeting at each vertex.
Alternative names: Rectified 120-cell (Norman Johnson) Rectified hecatonicosichoron / rectified dodecacontachoron / rectified polydodecahedron Icosidodecahedral hexacosihecatonicosachoron Rahi (Jonathan Bowers: for rectified hecatonicosachoron) Ambohecatonicosachoron (Neil Sloane & John Horton Conway)
Projections 3D parallel projectionParallel projection of the rectified 120-cell into 3D, centered on an icosidodecahedral cell. Nearest cell to 4D viewpoint shown in orange, and tetrahedral cells shown in yellow. Remaining cells culled so that the structure of the projection is visible.
+ Orthographic projections by Coxeter planesH4 -F4H3A2 / B3 / D4A3 / B2
Related polytopes
Notes
References Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, (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] J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
rectified 120-cell Marco Möller's Archimedean polytopes in R4 (German)
Four-dimensional Archimedean Polytopes, Marco Möller, 2004 PhD dissertation