Rectified 600-cell

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Rectified 600-cell
Rectified 600-cell schlegel halfsolid.png
Schlegel diagram, shown as Birectified 120-cell, with 119 icosahedral cells colored
Type Uniform 4-polytope
Uniform index 34
Schläfli symbol t1{3,3,5}
or r{3,3,5}
Coxeter-Dynkin diagram CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 600 (3.3.3.3) Uniform polyhedron-33-t1.png
120 {3,5} Icosahedron.png
Faces 1200+2400 {3}
Edges 3600
Vertices 720
Vertex figure Rectified 600-cell verf.png
pentagonal prism
Symmetry group H4, [3,3,5], order 14400
Properties convex, vertex-transitive, edge-transitive

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.

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[edit]

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[edit]

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

Images[edit]

Orthographic projections by Coxeter planes
H4 - F4
600-cell t1 H4.svg
[30]
600-cell t1 p20.svg
[20]
600-cell t1 F4.svg
[12]
H3 A2 / B3 / D4 A3 / B2
600-cell t1 H3.svg
[10]
600-cell t1 A2.svg
[6]
600-cell t1.svg
[4]
Stereographic projection Net
Stereographic rectified 600-cell.png Rectified hexacosichoron net.png

Related polytopes[edit]

Diminished rectified 600-cell[edit]

120-diminished rectified 600-cell
Type 4-polytope
Cells 840 cells:
600 square pyramid
120 pentagonal prism
120 pentagonal antiprism
Faces 2640:
1800 {3}
600 {4}
240 {5}
Edges 2400
Vertices 600
Vertex figure Spidrox-vertex figure.png
Bi-diminished pentagonal prism
(1) 3.3.3.3 + (4) 3.3.4 Square pyramid.png
(2) 4.4.5 Pentagonal prism.png
(2) 3.3.3.5 Pentagonal antiprism.png
Symmetry group 1/12[3,3,5], order 1200
Properties convex

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 pentagaonal 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
Spidrox-ring2-perspective.png
Two orthogonal rings shown
Spidrox-square pyramid ring.png
2 rings of 30 red square pyramids, one ring along perimeter, and one centered.

Swirlprismatodiminished rectified hexacosichoron net.png
Net

H4 family[edit]

Pentagonal prism vertex figures[edit]

r{p,3,5}
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name r{3,3,5}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
r{4,3,5}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
r{5,3,5}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
r{6,3,5}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
CDel branch 11.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
r{7,3,5}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
... r{∞,3,5}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
Image Stereographic rectified 600-cell.png H3 435 CC center 0100.png H3 535 CC center 0100.png H3 635 boundary 0100.png
Cells
Icosahedron.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform polyhedron-33-t1.png
r{3,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cuboctahedron.png
r{4,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.png
r{5,3}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.svg
r{6,3}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 237-2.png
r{7,3}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 23i-2.png
r{∞,3}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png

References[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
  • Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [2]

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / 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