Rectified 120-cell

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Four rectifications
120-cell t0 H3.svg
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
120-cell t1 H3.svg
Rectified 120-cell
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
600-cell t0 H3.svg
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
600-cell t1 H3.svg
Rectified 600-cell
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

Rectified 120-cell
Rectified 120-cell schlegel halfsolid.png
Schlegel diagram, centered on icosidodecahedon, tetrahedral cells visible
Type Uniform 4-polytope
Uniform index 33
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schläfli symbol t1{5,3,3}
or r{5,3,3}
Cells 720 total:
120 ( Icosidodecahedron.png
600 (3.3.3) Tetrahedron.png
Faces 3120 total:
2400 {3}, 720 {5}
Edges 3600
Vertices 1200
Vertex figure Rectified 120-cell verf.png
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)


3D parallel projection
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
120-cell t1 H4.svg
120-cell t1 p20.svg
120-cell t1 F4.svg
H3 A2 / B3 / D4 A3 / B2
120-cell t1 H3.svg
120-cell t1 A2.svg
120-cell t1 A3.svg

Related polytopes[edit]



  • 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, ISBN 978-0-471-01003-6 [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[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds