Rectified 120-cell

Four rectifications

File:120-cell t0 H3.svg 120-cell Template:CDD	File:120-cell t1 H3.svg Rectified 120-cell Template:CDD
File:600-cell t0 H3.svg 600-cell Template:CDD	File:600-cell t1 H3.svg Rectified 600-cell Template:CDD
Orthogonal 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-cell
File:Rectified 120-cell schlegel halfsolid.png Schlegel diagram, centered on icosidodecahedon, tetrahedral cells visible
Type	Uniform 4-polytope
Uniform index	33
Coxeter diagram	Template:CDD
Schläfli symbol	t1{5,3,3} or r{5,3,3}
Cells	720 total: 120 (3.5.3.5) File:Icosidodecahedron.png 600 (3.3.3) File:Tetrahedron.png
Faces	3120 total: 2400 {3}, 720 {5}
Edges	3600
Vertices	1200
Vertex figure	File:Rectified 120-cell verf.png triangular prism
Symmetry group	H4 or [3,3,5]
Properties	convex, vertex-transitive, edge-transitive

File:Rectified hecatonicosachoron net.png

Net

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 projection
File:Rectified 120-cell-parallel-cell-first-01.png	Parallel 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 planes

H4	-	F4
File:120-cell t1 H4.svg [30]	File:120-cell t1 p20.svg [20]	File:120-cell t1 F4.svg [12]
H3	A2 / B3 / D4	A3 / B2
File:120-cell t1 H3.svg [10]	File:120-cell t1 A2.svg [6]	File:120-cell t1 A3.svg [4]

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, Template:Isbn [1]
  • (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

• Template:PolyCell
• rectified 120-cell Marco Möller's Archimedean polytopes in R⁴ (German)
• Template:KlitzingPolytopes
• Template:In lang Four-dimensional Archimedean Polytopes, Marco Möller, 2004 PhD dissertation [2]
• H4 uniform polytopes with coordinates: r{5,3,3}