Rectified 600-cell

In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.

Rectified 600-cell

Schlegel diagram, shown as Birectified 120-cell, with 119 icosahedral cells colored
TypeUniform 4-polytope
Uniform index34
Schläfli symbolt1{3,3,5}
or r{3,3,5}
Coxeter-Dynkin diagram
Cells600 (3.3.3.3)
120 {3,5}
Faces1200+2400 {3}
Edges3600
Vertices720
Vertex figure
pentagonal prism
Symmetry groupH4, [3,3,5], order 14400
Propertiesconvex, vertex-transitive, edge-transitive

Containing the cell realms of both the regular 120-cell and the regular 600-cell, it can be considered analogous to the polyhedron icosidodecahedron, which is a rectified icosahedron and rectified dodecahedron.

The vertex figure of the rectified 600-cell is a uniform pentagonal prism.

Semiregular polytope

It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a octicosahedric for being made of octahedron and icosahedron cells.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC600.

Alternate names
  • octicosahedric (Thorold Gosset)
  • Icosahedral hexacosihecatonicosachoron
  • Rectified 600-cell (Norman W. Johnson)
  • Rectified hexacosichoron
  • Rectified polytetrahedron
  • Rox (Jonathan Bowers)

Images
Orthographic projections by Coxeter planes
H4 - F4

[30]
[20]
[12]
H3 A2 / B3 / D4 A3 / B2

[10]
[6]
[4]
Stereographic projection Net

Diminished rectified 600-cell
120-diminished rectified 600-cell
Type4-polytope
Cells840 cells:
600 square pyramid
120 pentagonal prism
120 pentagonal antiprism
Faces2640:
1800 {3}
600 {4}
240 {5}
Edges2400
Vertices600
Vertex figure
Bi-diminished pentagonal prism
(1) 3.3.3.3 + (4) 3.3.4
(2) 4.4.5
(2) 3.3.3.5
Symmetry group1/12[3,3,5], order 1200
Propertiesconvex

A related vertex-transitive polytope can be constructed with equal edge lengths removes 120 vertices from the rectified 600-cell, but isn't uniform because it contains square pyramid cells,[1] discovered by George Olshevsky, calling it a swirlprismatodiminished rectified hexacosichoron, with 840 cells (600 square pyramids, 120 pentagonal prisms, and 120 pentagonal antiprisms), 2640 faces (1800 triangles, 600 square, and 240 pentagons), 2400 edges, and 600 vertices. It has a chiral bi-diminished pentagonal prism vertex figure.

Each removed vertex creates a pentagonal prism cell, and diminishes two neighboring icosahedra into pentagonal antiprisms, and each octahedron into a square pyramid.[2]

This polytope can be partitioned into 12 rings of alternating 10 pentagonal prisms and 10 antiprisms, and 30 rings of square pyramids.

Schlegel diagram Orthogonal projection

Two orthogonal rings shown
2 rings of 30 red square pyramids, one ring along perimeter, and one centered.


Net

H4 family

Pentagonal prism vertex figures
r{p,3,5}
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name r{3,3,5}
 r{4,3,5}

 r{5,3,5}
 r{6,3,5}

 r{7,3,5}
 ... r{∞,3,5}

Image
Cells

{3,5}
r{3,3}
r{4,3}
r{5,3}
r{6,3}
r{7,3}
r{∞,3}

References
  1. Category S4: Scaliform Swirlprisms spidrox
  2. Klitzing, Richard. "4D convex scaliform polychora swirlprismatodiminished rectified hexacosachoron".
  • 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
    • (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
  • Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation
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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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