Rectified 600-cell

Summary

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 polytopeEdit

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 namesEdit

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

ImagesEdit

Orthographic projections by Coxeter planes
H4 - F4
 
[30]
 
[20]
 
[12]
H3 A2 / B3 / D4 A3 / B2
 
[10]
 
[6]
 
[4]
Stereographic projection Net
   

Related polytopesEdit

Diminished rectified 600-cellEdit

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  
Bi-diminished pentagonal prism
(1) 3.3.3.3 + (4) 3.3.4  
(2) 4.4.5  
(2) 3.3.3.5  
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 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 familyEdit

H4 family polytopes
120-cell rectified
120-cell
truncated
120-cell
cantellated
120-cell
runcinated
120-cell
cantitruncated
120-cell
runcitruncated
120-cell
omnitruncated
120-cell
                                                               
{5,3,3} r{5,3,3} t{5,3,3} rr{5,3,3} t0,3{5,3,3} tr{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
               
             
600-cell rectified
600-cell
truncated
600-cell
cantellated
600-cell
bitruncated
600-cell
cantitruncated
600-cell
runcitruncated
600-cell
omnitruncated
600-cell
                                                               
{3,3,5} r{3,3,5} t{3,3,5} rr{3,3,5} 2t{3,3,5} tr{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}

Pentagonal prism vertex figuresEdit

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}
     

ReferencesEdit

  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 [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 linksEdit

  • Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 34, George Olshevsky.
  • Klitzing, Richard. "4D uniform polytopes (polychora) o3x3o5o - rox".
  • Archimedisches Polychor Nr. 45 (rectified 600-cell) Marco Möller's Archimedean polytopes in R4 (German)
  • H4 uniform polytopes with coordinates: r{3,3,5}
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 polychoron Pentachoron 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