View source for Rectified 600-cell

{|class="wikitable" style="float:right; margin-left:8px; width:250px"
!bgcolor=#e7dcc3 colspan=2|Rectified 600-cell
|-
|bgcolor=#ffffff align=center colspan=2|[[File:Rectified 600-cell schlegel halfsolid.png|280px]]<br>[[Schlegel diagram]], shown as Birectified [[120-cell]], with 119 icosahedral cells colored
|-
|bgcolor=#e7dcc3|Type||[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|Uniform index||34
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3,5}<br>or r{3,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node|5|node|3|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells||600 ([[Octahedron|''3.3.3.3'']]) [[Image:Uniform polyhedron-33-t1.svg|20px]]<br>120 [[Icosahedron|{3,5}]] [[Image:Icosahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces||1200+2400 {3}
|-
|bgcolor=#e7dcc3|Edges||3600
|-
|bgcolor=#e7dcc3|Vertices||720
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Rectified 600-cell verf.png|80px]]<br>[[pentagonal prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]||H<sub>4</sub>, [3,3,5], order 14400
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[vertex-transitive]], [[edge-transitive]]
|}
In [[geometry]], the '''[[Rectification (geometry)|rectified]] [[600-cell]]''' or '''rectified hexacosichoron''' is a convex [[uniform 4-polytope]] composed of 600 regular octahedra and 120 icosahedra [[cell (mathematics)|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 [[Hyperplane#Notes|realm]]s 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#Semiregular polytopes|semiregular 4-polytopes]] made of two or more cells which are [[Platonic solid]]s, discovered by [[Thorold Gosset]] in his 1900 paper. He called it a ''octicosahedric'' for being made of [[octahedron]] and [[icosahedron]] cells.

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as tC<sub>600</sub>.

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

== Images ==

{|class="wikitable"
|+ [[Orthographic projection]]s by [[Coxeter plane]]s
|- align=center
!H<sub>4</sub>
! -
!F<sub>4</sub>
|- align=center
|[[File:600-cell t1 H4.svg|160px]]<br>[30]
|[[File:600-cell t1 p20.svg|160px]]<br>[20]
|[[File:600-cell t1 F4.svg|160px]]<br>[12]
|- align=center
!H<sub>3</sub>
!A<sub>2</sub> / B<sub>3</sub> / D<sub>4</sub>
!A<sub>3</sub> / B<sub>2</sub>
|- align=center
|[[File:600-cell t1 H3.svg|160px]]<br>[10]
|[[File:600-cell t1 A2.svg|160px]]<br>[6]
|[[File:600-cell t1.svg|160px]]<br>[4]
|}

{|class=wikitable
![[Stereographic projection]]
![[Net (polyhedron)|Net]]
|-
|[[Image:Stereographic rectified 600-cell.png|220px]]
|[[File:Rectified hexacosichoron net.png|220px]]
|}

== Related polytopes ==

=== Diminished rectified 600-cell ===
{|class="wikitable" style="float:right; margin-left:8px; width:250px"
!bgcolor=#e7dcc3 colspan=2|120-diminished rectified 600-cell
|-
|bgcolor=#e7dcc3|Type||4-polytope
|-
|bgcolor=#e7dcc3|Cells||840 cells:<BR>600 [[square pyramid]]<BR>120 [[pentagonal prism]]<BR>120 [[pentagonal antiprism]]
|-
|bgcolor=#e7dcc3|Faces||2640:<BR>1800 {3}<BR>600 {4}<BR>240 {5}
|-
|bgcolor=#e7dcc3|Edges||2400
|-
|bgcolor=#e7dcc3|Vertices||600
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Spidrox-vertex figure.png|80px]]<br>Bi-diminished [[pentagonal prism]]<BR>(1) [[square pyramid|3.3.3.3]] + (4) [[square pyramid|3.3.4]] [[File:Square pyramid.png|20px]]<br>(2) [[pentagonal prism|4.4.5]] [[File:Pentagonal prism.png|20px]]<br>(2) [[pentagonal antiprism|3.3.3.5]] [[File:Pentagonal antiprism.png|20px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]||1/12[3,3,5], order 1200
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|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,<ref>[http://www.polytope.net/hedrondude/scaleswirl.htm Category S4: Scaliform Swirlprisms] spidrox</ref> 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.<ref>{{KlitzingPolytopes|../incmats/spidrox.htm|4D convex scaliform polychora|swirlprismatodiminished rectified hexacosachoron}}</ref>

This polytope can be partitioned into 12 rings of alternating 10 pentagonal prisms and 10 antiprisms, and 30 rings of square pyramids.
{| class=wikitable width=360
![[Schlegel diagram]]
!Orthogonal projection
|- align=center valign=top
|[[File:Spidrox-ring2-perspective.png|180px]]<br>Two orthogonal rings shown
|[[File:Spidrox-square pyramid ring.png|180px]]<br>2 rings of 30 red square pyramids, one ring along perimeter, and one centered.
|}

[[File:Swirlprismatodiminished rectified hexacosichoron net.png|200px]]<BR>Net
{{-}}

=== H4 family ===
{{H4_family}}

=== Pentagonal prism vertex figures ===
{{Pentagonal prism vertex figure tessellations}}

== References ==
{{reflist}}
* '''Kaleidoscopes: Selected Writings of [[Harold Scott MacDonald Coxeter|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]
* [[John Horton Conway|J.H. Conway]] and [[Michael Guy (computer scientist)|M.J.T. Guy]]: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
* [[Norman Johnson (mathematician)|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 R<sup>4</sup> (German)
* H4 uniform polytopes with coordinates: [http://eusebeia.dyndns.org/4d/rect600cell r{3,3,5}]

{{Polytopes}}

[[Category:Uniform 4-polytopes]]