{{Short description|Regular star 4-polytope with 600 faces}} {{Noinline|date=May 2023}} {| class="wikitable" align="right" style="margin-left:10px" width="250" !bgcolor=#e7dcc3 colspan=2|Grand 600-cell |- |bgcolor=#ffffff align=center colspan=2|280px<BR>Orthogonal projection |- |bgcolor=#e7dcc3|Type||Regular star 4-polytope |- |bgcolor=#e7dcc3|Cells||600 {3,3} |- |bgcolor=#e7dcc3|Faces||1200 {3} |- |bgcolor=#e7dcc3|Edges||720 |- |bgcolor=#e7dcc3|Vertices||120 |- |bgcolor=#e7dcc3|Vertex figure||{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|Symmetry group||H<sub>4</sub>, [3,3,5] |- |bgcolor=#e7dcc3|Dual|| Great grand stellated 120-cell |- |bgcolor=#e7dcc3|Properties|| Regular |} In 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 Ludwig Schläfli. It was named by John Horton Conway, extending the naming system by Arthur Cayley for the Kepler-Poinsot solids.

The grand 600-cell can be seen as the four-dimensional analogue of the great icosahedron (which in turn is analogous to the pentagram); both of these are the only regular ''n''-dimensional star polytopes which are derived by performing stellational operations on the pentagonal polytope which has 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 triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces.

The Grand 600-cell is also dual to the great grand stellated 120-cell, mirroring the great icosahedron's duality with the 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, and grand stellated 120-cell, and same face arrangement as the great icosahedral 120-cell.

{| class="wikitable" width=600 |+ Orthographic projections 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 |200px |200px |200px |}

== See also == * List of regular polytopes * Convex regular 4-polytope * Kepler-Poinsot solids - regular 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}}. * 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] {{Webarchive|url=https://web.archive.org/web/20221217061349/https://homepages.wmich.edu/~drichter/great600cell.htm |date=2022-12-17 }} {{sic}}

{{Regular 4-polytopes}}

Category:Regular 4-polytopes {{polychora-stub}}