View source for Grand 600-cell

{{Short description|Regular star 4-polytope with 600 faces}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Grand 600-cell
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Ortho solid 015-uniform polychoron 33p-t0.png|280px]]<BR>Orthogonal projection
|-
|bgcolor=#e7dcc3|Type||Regular star 4-polytope
|-
|bgcolor=#e7dcc3|Cells||600 {3,3}
|-
|bgcolor=#e7dcc3|Faces||1200 [[Triangle|{3}]]
|-
|bgcolor=#e7dcc3|Edges||720
|-
|bgcolor=#e7dcc3|Vertices||120
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Great icosahedron|{3,5/2}]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,5/2}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|node_1|3|node|3|node|5|rat|d2|node}}
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]||H<sub>4</sub>, [3,3,5]
|-
|bgcolor=#e7dcc3|Dual|| [[Great grand stellated 120-cell]]
|-
|bgcolor=#e7dcc3|Properties|| Regular
|}
In [[Geometry|geometry]], the '''grand 600-cell''' or '''grand polytetrahedron''' is a regular star 4-polytope with [[Schläfli symbol]] {3, 3, 5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells.

It is one of four ''regular star 4-polytopes'' discovered by [[Biography:Ludwig Schläfli|Ludwig Schläfli]]. It was named by [[Biography:John Horton Conway|John Horton Conway]], extending the naming system by [[Biography:Arthur Cayley|Arthur Cayley]] for the Kepler-Poinsot solids.

The grand 600-cell can be seen as the four-dimensional analogue of the [[Great icosahedron|great icosahedron]] (which in turn is analogous to the [[Pentagram|pentagram]]); both of these are the only regular ''n''-dimensional star polytopes which are derived by performing stellational operations on the [[Pentagonal polytope|pentagonal polytope]] which has [[Simplex|simplectic]] faces. It can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of said (''n-1'')-D simplex faces of the core ''n''D polytope (tetrahedra for the grand 600-cell, [[Equilateral triangle|equilateral triangle]]s for the great icosahedron, and [[Line segment|line segment]]s for the pentagram) until the figure regains regular faces.

The Grand 600-cell is also dual to the [[Great grand stellated 120-cell|great grand stellated 120-cell]], mirroring the great icosahedron's duality with the [[Great stellated dodecahedron|great stellated dodecahedron]] (which in turn is also analogous to the pentagram); all of these are the final stellations of the ''n''-dimensional "dodecahedral-type" pentagonal polytope.

== Related polytopes ==

It has the same edge arrangement as the [[Great stellated 120-cell|great stellated 120-cell]], and [[Grand stellated 120-cell|grand stellated 120-cell]], and same face arrangement as the [[Great icosahedral 120-cell|great icosahedral 120-cell]].

{| class="wikitable" width=600
|+ [[Orthographic projection]]s by Coxeter planes
|- 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:Schläfli-Hess_polychoron-wireframe-4.png|200px]]
|[[File:Grand_600-cell-ortho-6gon.png|200px]]
|[[File:Grand_600-cell-ortho-4gon.png|200px]]
|}

== See also ==
* List of regular polytopes
* Convex regular 4-polytope
* 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)|x3o3o5/2o - gax}}

== 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/great600cell.htm The Great 600-cell, a Zome Model] {{sic}}
{{Regular 4-polytopes}}
[[Category:4-polytopes]]

{{Sourceattribution|Grand 600-cell}}