Rectified 600-cell

Rectified 600-cell
Four rectifications
600-cell t0 H3.svg
600-cell
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
120-cell t03 H3.png
Rectified 120-cell
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
120-cell t0 H3.svg
120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Orthogonal projections in H3 Coxeter plane

In geometry, a rectified 600-cell is a uniform polychoron (4-dimensional uniform polytope) formed as the rectification of the regular 600-cell.

There are four rectifications of the 600-cell, including the zeroth, the 600-cell itself. Tbe birectified 600-cell is more easily seen as a rectified 120-cell, and the trirectified 600-cell is the same as the dual 120-cell.

Contents


Rectified 600-cell

Rectified 600-cell
Rectified 600-cell schlegel halfsolid.png
Schlegel diagram, shown as Birectified 120-cell, with 119 icosahedral cells colored
Type Uniform polychoron
Uniform index 34
Schläfli symbol t1{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 or [3,3,5]
Properties convex, edge-transitive
Vertex figure: pentagonal prism
7 faces:
Octahedron vertfig.pngIcosahedron vertfig.png
5 (3.3.3.3) and 2 (3.3.3.3.3)

In geometry, the rectified 600-cell is a convex uniform polychoron 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.

It is one of three semiregular polychora 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.

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.

Alternate names

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


Images

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 A3.svg
[4]
Stereographic projection
Stereographic rectified 600-cell.png


Related polytopes

H4 family polytopes by name, Coxeter-Dynkin diagram, and Schläfli symbol
120-cell rectified
120-cell
truncated
120-cell
cantellated
120-cell
runcinated
120-cell
bitruncated
120-cell
cantitruncated
120-cell
runcitruncated
120-cell
omnitruncated
120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
{5,3,3} t1{5,3,3} t0,1{5,3,3} t0,2{5,3,3} t0,3{5,3,3} t1,2{5,3,3} t0,1,2{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
120-cell t0 H3.svg 120-cell t1 H3.svg 120-cell t01 H3.svg 120-cell t02 H3.png 120-cell t03 H3.png 120-cell t12 H3.png 120-cell t012 H3.png 120-cell t013 H3.png 120-cell t0123 H3.png
600-cell t0 H3.svg 600-cell t1 H3.svg 600-cell t01 H3.svg 600-cell t02 H3.svg 120-cell t03 H3.png 120-cell t023 H3.png
600-cell rectified
600-cell
truncated
600-cell
cantellated
600-cell
runcinated
600-cell
bitruncated
600-cell
cantitruncated
600-cell
runcitruncated
600-cell
omnitruncated
600-cell
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
{3,3,5} t1{3,3,5} t0,1{3,3,5} t0,2{3,3,5} t0,3{3,3,5} t1,2{3,3,5} t0,1,2{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}

References

  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied 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


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