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Rectified 600-cell#Diminished rectified 600-cell

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!bgcolor=#e7dcc3 colspan=2|Rectified 600-cell

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Schlegel diagram, shown as Birectified 120-cell, with 119 icosahedral cells colored
bgcolor=#e7dcc3|TypeUniform 4-polytope
bgcolor=#e7dcc3|Uniform index34
bgcolor=#e7dcc3|Schläfli symbolt1{3,3,5}
or r{3,3,5}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node|5|node|3|node_1|3|node}}
bgcolor=#e7dcc3|Cells600 (3.3.3.3) 20px
120 {3,5} 20px
bgcolor=#e7dcc3|Faces1200+2400 {3}
bgcolor=#e7dcc3|Edges3600
bgcolor=#e7dcc3|Vertices720
bgcolor=#e7dcc3|Vertex figure80px
pentagonal prism
bgcolor=#e7dcc3|Symmetry groupH4, [3,3,5], order 14400
bgcolor=#e7dcc3|Propertiesconvex, 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

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

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|+ Orthographic projections by Coxeter planes

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!H4

! -

!F4

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|160px
[30]

|160px
[20]

|160px
[12]

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!H3

!A2 / B3 / D4

!A3 / B2

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|160px
[10]

|160px
[6]

|160px
[4]

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!Stereographic projection

!Net

220px

|220px

Related polytopes

= Diminished rectified 600-cell =

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!bgcolor=#e7dcc3 colspan=2|120-diminished rectified 600-cell

bgcolor=#e7dcc3|Type4-polytope
bgcolor=#e7dcc3|Cells840 cells:
600 square pyramid
120 pentagonal prism
120 pentagonal antiprism
bgcolor=#e7dcc3|Faces2640:
1800 {3}
600 {4}
240 {5}
bgcolor=#e7dcc3|Edges2400
bgcolor=#e7dcc3|Vertices600
bgcolor=#e7dcc3|Vertex figure80px
Bi-diminished pentagonal prism
(1) 3.3.3.3 + (4) 3.3.4 20px
(2) 4.4.5 20px
(2) 3.3.3.5 20px
bgcolor=#e7dcc3|Symmetry group1/12[3,3,5], order 1200
bgcolor=#e7dcc3|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,[http://www.polytope.net/hedrondude/scaleswirl.htm Category S4: Scaliform Swirlprisms] spidrox 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.{{KlitzingPolytopes|../incmats/spidrox.htm|4D convex scaliform polychora|swirlprismatodiminished rectified hexacosachoron}}

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

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!Schlegel diagram

!Orthogonal projection

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Two orthogonal rings shown

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

200px
Net

{{-}}

= H4 family =

{{H4_family}}

= Pentagonal prism vertex figures =

{{Pentagonal prism vertex figure tessellations}}

References

{{reflist}}

  • 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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{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
  • [http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf]

External links

  • {{PolyCell | urlname = section4.html| title = Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 34}}
  • {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|o3x3o5o - rox}}
  • [http://www.polytope.de/nr45.html Archimedisches Polychor Nr. 45 (rectified 600-cell)] Marco Möller's Archimedean polytopes in R4 (German)
  • H4 uniform polytopes with coordinates: [http://eusebeia.dyndns.org/4d/rect600cell r{3,3,5}]

{{Polytopes}}

Category:Uniform 4-polytopes