# Grand 600-cell
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> Source: https://en.wikipedia.org/wiki/Grand_600-cell
> Source revision: 1236331104
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{{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
[Orthogonal projection](/source/Orthogonal_projection)
|-
|bgcolor=#e7dcc3|Type||[Regular star 4-polytope](/source/Regular_star_4-polytope)
|-
|bgcolor=#e7dcc3|Cells||600 [{3,3}](/source/Tetrahedra)
|-
|bgcolor=#e7dcc3|Faces||1200 [{3}](/source/Triangle)
|-
|bgcolor=#e7dcc3|Edges||720
|-
|bgcolor=#e7dcc3|Vertices||120
|-
|bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||[{3,5/2}](/source/Great_icosahedron)
|-
|bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)|| {3,3,5/2}
|-
|bgcolor=#e7dcc3|[Coxeter-Dynkin diagram](/source/Coxeter-Dynkin_diagram)||{{CDD|node_1|3|node|3|node|5|rat|d2|node}}
|-
|bgcolor=#e7dcc3|[Symmetry group](/source/Coxeter_group)||H4, [3,3,5]
|-
|bgcolor=#e7dcc3|Dual|| [Great grand stellated 120-cell](/source/Great_grand_stellated_120-cell)
|-
|bgcolor=#e7dcc3|Properties|| Regular
|}
In [geometry](/source/geometry), the '''grand 600-cell''' or '''grand polytetrahedron''' is a [regular star 4-polytope](/source/regular_star_4-polytope) with [Schläfli symbol](/source/Schl%C3%A4fli_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](/source/Ludwig_Schl%C3%A4fli). It was named by [John Horton Conway](/source/John_Horton_Conway), extending the naming system by [Arthur Cayley](/source/Arthur_Cayley) for the [Kepler-Poinsot solid](/source/Kepler-Poinsot_solid)s.
The grand 600-cell can be seen as the four-dimensional analogue of the [great icosahedron](/source/great_icosahedron) (which in turn is analogous to the [pentagram](/source/pentagram)); both of these are the only regular ''n''-dimensional star polytopes which are derived by performing stellational operations on the [pentagonal polytope](/source/pentagonal_polytope) which has [simplectic](/source/simplex) 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](/source/tetrahedra) for the grand 600-cell, [equilateral triangle](/source/equilateral_triangle)s for the great icosahedron, and [line segment](/source/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](/source/great_grand_stellated_120-cell), mirroring the great icosahedron's duality with the [great stellated dodecahedron](/source/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](/source/edge_arrangement) as the [great stellated 120-cell](/source/great_stellated_120-cell), and [grand stellated 120-cell](/source/grand_stellated_120-cell), and same [face arrangement](/source/face_arrangement) as the [great icosahedral 120-cell](/source/great_icosahedral_120-cell).
{| class="wikitable" width=600
|+ [Orthographic projection](/source/Orthographic_projection)s by [Coxeter plane](/source/Coxeter_plane)s
|- align=center
!H3
!A2 / B3 / D4
!A3 / B2
|- align=center
|200px
|200px
|200px
|}
== See also ==
* [List of regular polytopes](/source/List_of_regular_polytopes)
* [Convex regular 4-polytope](/source/Convex_regular_4-polytope)
* [Kepler-Poinsot solid](/source/Kepler-Poinsot_solid)s - regular [star polyhedron](/source/star_polyhedron)
* [Star polygon](/source/Star_polygon) - regular star polygons
== References ==
* [Edmund Hess](/source/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](/source/Coxeter), ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. {{ISBN|0-486-61480-8}}.
* [John H. Conway](/source/John_Horton_Conway), Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26, Regular Star-polytopes, pp. 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}}
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Adapted from the Wikipedia article [Grand 600-cell](https://en.wikipedia.org/wiki/Grand_600-cell) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Grand_600-cell?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.