Rectified 120-cell explained

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

bgcolor=#e7dcc3 colspan=2	Rectified 120-cell
bgcolor=#ffffff align=center colspan=2	Schlegel diagram, centered on icosidodecahedon, tetrahedral cells visible
Type	Uniform 4-polytope
Uniform index	33
Coxeter diagram
Schläfli symbol	t1 or r
Cells
Faces	3120 total: 2400 , 720
Edges	3600
Vertices	1200
Vertex figure	triangular prism
Symmetry group	H4 or [3,3,5]
Properties	convex, 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)

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (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 http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf
• H4 uniform polytopes with coordinates: r