# Rectified 120-cell

 Orthogonal projections in H3 Coxeter plane 120-cell Rectified 120-cell 600-cell Rectified 600-cell

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

Schlegel diagram, centered on icosidodecahedon, tetrahedral cells visible
Type Uniform 4-polytope
Uniform index 33
Coxeter diagram
Schläfli symbol t1{5,3,3}
or r{5,3,3}
Cells 720 total:
120 (3.5.3.5)
600 (3.3.3)
Faces 3120 total:
2400 {3}, 720 {5}
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)

## Projections

3D parallel projection
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

[30]

[20]

[12]
H3 A2 / B3 / D4 A3 / B2

[10]

[6]

[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, 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