View source for Great grand stellated 120-cell

{{Short description|Regular Schläfli-Hess 4-polytope with 600 vertices}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Great grand stellated 120-cell
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Ortho solid 016-uniform polychoron p33-t0.png|280px]]<BR>Orthogonal projection
|-
|bgcolor=#e7dcc3|Type||Schläfli-Hess polychoron
|-
|bgcolor=#e7dcc3|Cells||120 [[Great stellated dodecahedron|{5/2,3}]]
|-
|bgcolor=#e7dcc3|Faces||720 [[Pentagram|{5/2}]]
|-
|bgcolor=#e7dcc3|Edges||1200
|-
|bgcolor=#e7dcc3|Vertices||600
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Tetrahedron|{3,3}]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {5/2,3,3}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|node_1|5|rat|d2|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]||H<sub>4</sub>, [3,3,5]
|-
|bgcolor=#e7dcc3|Dual|| [[Grand 600-cell]]
|-
|bgcolor=#e7dcc3|Properties|| Regular
|}
[[Image:Great grand stellated 120cell zome 032407.jpg|thumb|A [[Engineering:Zome|Zome]] model]]
In [[Geometry|geometry]], the '''great grand stellated 120-cell''' or '''great grand stellated polydodecahedron''' is a regular star 4-polytope with [[Schläfli symbol]] {5/2,3,3}, one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and has the same [[Vertex arrangement|vertex arrangement]] as the regular convex [[120-cell|120-cell]].

It is one of four ''regular star polychora'' discovered by [[Biography:Ludwig Schläfli|Ludwig Schläfli]]. It is named by [[Biography:John Horton Conway|John Horton Conway]], extending the naming system by [[Biography:Arthur Cayley|Arthur Cayley]] for the Kepler-Poinsot solids, and the only one containing all three modifiers in the name.

With its dual, it forms the compound of great grand stellated 120-cell and grand 600-cell.

==Images==
{| class=wikitable
|+ Coxeter plane images
!H<sub>4</sub>
!A<sub>2</sub> / B<sub>3</sub>
!A<sub>3</sub> / B<sub>2</sub>
|-
!colspan=3|Great grand stellated 120-cell, {5/2,3,3}
|-
|[[File:Great_grand_stellated_120-cell-ortho-10gon.png|200px]]
|[[File:Great_grand_stellated_120-cell-6gon.png|200px]]
|[[File:Great_grand_stellated_120-cell-4gon.png|200px]]
|-
![10]
![6]
![4]
|-
!colspan=3|120-cell, {5,3,3}
|-
|[[File:120-cell t0 H3.svg|210px]]
|[[File:120-cell t0 A2.svg|210px]]
|[[File:120-cell t0 A3.svg|210px]]
|}

== As a stellation ==

The '''great grand stellated 120-cell''' is the ''final [[Stellation|stellation]]'' of the [[120-cell|120-cell]], and is the only Schläfli-Hess polychoron to have the 120-cell for its convex hull. In this sense it is analogous to the three-dimensional [[Great stellated dodecahedron|great stellated dodecahedron]], which is the final stellation of the [[Dodecahedron|dodecahedron]] and the only Kepler-Poinsot polyhedron to have the dodecahedron for its convex hull. Indeed, the great grand stellated 120-cell is dual to the [[Grand 600-cell|grand 600-cell]], which could be taken as a 4D analogue of the [[Great icosahedron|great icosahedron]], dual of the great stellated dodecahedron.

The edges of the great grand stellated 120-cell are [[Golden ratio|τ]]<sup>6</sup> as long as those of the 120-cell core deep inside the polychoron, and they are τ<sup>3</sup> as long as those of the [[Small stellated 120-cell|small stellated 120-cell]] deep within the polychoron.

== See also ==
* List of regular polytopes
* Convex regular 4-polytope – Set of convex regular polychora
* Kepler-Poinsot solids – regular [[Star polyhedron|star polyhedron]]
* [[Star polygon]] – regular star polygons

== References ==
* Edmund Hess, (1883) ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder'' [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001].
*H. S. M. Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. {{ISBN|0-486-61480-8}}.
* [[Biography:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26, Regular Star-polytopes, pp.&nbsp;404–408)
* {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|o3o3o5/2x - gogishi}}

== External links ==
* [http://hometown.aol.com/hedrondude/regulars.html Regular polychora] {{Webarchive|url=https://web.archive.org/web/20030906012615/http://hometown.aol.com/hedrondude/regulars.html |date=2003-09-06 }}
* [http://mathforum.org/library/drmath/view/54786.html Discussion on names]
* [https://web.archive.org/web/20061107052613/http://www.mathematik.uni-regensburg.de/Goette/sterne/ Reguläre Polytope]
* [https://web.archive.org/web/20070704012333/http://davidf.faricy.net/polyhedra/Star_Polychora.html The Regular Star Polychora]
* [http://homepages.wmich.edu/~drichter/finalstellation120cell.htm Zome Model of the Final Stellation of the 120-cell]
{{Regular 4-polytopes}}
{{Polytopes}}

[[Category:4-polytopes]]

{{Sourceattribution|Great grand stellated 120-cell}}