Icosahedral 120-cell: Difference between revisions

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Icosahedral 120-cell
Orthogonal projection
Type	Schläfli-Hess polytope
Cells	120 {3,5}
Faces	1200 {3}
Edges	720
Vertices	120
Vertex figure	{5,5/2}
Schläfli symbol	{3,5,5/2}
Symmetry group	H₄, [3,3,5]
Coxeter-Dynkin diagram
Dual	Small stellated 120-cell
Properties	Regular

In geometry, the icosahedral 120-cell, polyicosahedron, faceted 600-cell or icosaplex is a regular star 4-polytope with Schläfli symbol {3,5,5/2}. It is one of 10 regular Schläfli-Hess polytopes.

It is constructed by 5 icosahedra around each edge in a pentagrammic figure. The vertex figure is a great dodecahedron.

Related polytopes

It has the same edge arrangement as the 600-cell, grand 120-cell and great 120-cell, and shares its vertices with all other Schläfli–Hess 4-polytopes except the great grand stellated 120-cell (another stellation of the 120-cell).

Orthographic projections by Coxeter planes
H₄	-	F₄
[30]	[20]	[12]
H₃	A₂ / B₃ / D₄	A₃ / B₂
[10]	[6]	[4]

As a faceted 600-cell, replacing the simplicial cells of the 600-cell with icosahedral pentagonal polytope cells, it could be seen as a four-dimensional analogue of the great dodecahedron, which replaces the triangular faces of the icosahedron with pentagonal faces. Indeed, the icosahedral 120-cell is dual to the small stellated 120-cell, which could be taken as a 4D analogue of the small stellated dodecahedron, dual of the great dodecahedron.

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 [1].
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. 404–408)
Klitzing, Richard. "4D uniform polytopes (polychora) x3o5o5/2o - fix".

External links

Regular polychora Archived 2003-09-06 at the Wayback Machine
Discussion on names
Reguläre Polytope
The Regular Star Polychora

This 4-polytope article is a stub. You can help Wikipedia by expanding it.

@@ Line 4: / Line 4: @@
 |bgcolor=#ffffff align=center colspan=2|[[Image:Ortho solid 007-uniform polychoron 35p-t0.png|280px]]<BR>[[Orthogonal projection]]
 |-
-|bgcolor=#e7dcc3|Type||[[Schläfli-Hess polychoron]]
+|bgcolor=#e7dcc3|Type||[[Schläfli-Hess polytope]]
 |-
 |bgcolor=#e7dcc3|Cells||120 [[Icosahedron|{3,5}]]
@@ Line 24: / Line 24: @@
 |bgcolor=#e7dcc3|Dual|| [[Small stellated 120-cell]]
 |-
-|bgcolor=#e7dcc3|Properties||--
+|bgcolor=#e7dcc3|Properties|| Regular
 |}
+In [[geometry]], the '''icosahedral 120-cell''', '''polyicosahedron''', '''[[faceting|faceted]] 600-cell''' or '''icosaplex''' is a regular [[star 4-polytope]] with [[Schläfli symbol]] {3,5,5/2}. It is one of 10 regular [[Schläfli-Hess polytope]]s.
-[[Image:Schläfli-Hess polychoron-wireframe-3.png|280px|thumb|[[Orthogonal projection]] as a wireframe]]
-In [[geometry]], the '''icosahedral 120-cell''' is a [[star polychoron]] with [[Schläfli symbol]] {3,5,5/2}. It is one of 10 regular [[Schläfli-Hess polychoron|Schläfli-Hess polychora]].
-It is constructed by 5 [[icosahedron|icosahedra]] around each edge in a [[pentagram|pentagrammic]] figure. The [[vertex figure]] is a [[great dodecahedron]].
+It is constructed by 5 [[icosahedron|icosahedra]] around each edge in a [[pentagram]]mic figure. The [[vertex figure]] is a [[great dodecahedron]].
 == Related polytopes ==
-It has the same [[edge arrangement]] as the [[600-cell]], [[grand 120-cell]] and [[great 120-cell]], and shares its vertices with each other Schlaffi-Hess Polychoron, besides the [[Great grand stellated 120-cell]], another stellation of the [[Hecatonicosachoron]].
+It has the same [[edge arrangement]] as the [[600-cell]], [[grand 120-cell]] and [[great 120-cell]], and shares its vertices with all other [[Schläfli–Hess 4-polytope]]s except the [[great grand stellated 120-cell]] (another stellation of the [[120-cell]]).
-[[Image:IcosahedralHecatonicosa.png|300px|Schlegel Diagram. Either cell-centered or vertex centered]]
+{| class="wikitable" width=600
-== References ==
+|+ [[Orthographic projection]]s by [[Coxeter plane]]s
-* [[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].
+|- align=center
-*[[Coxeter|H. S. M. Coxeter]], ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
+!H<sub>4</sub>
-* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404-408)
+! -
+!F<sub>4</sub>
+|- align=center
+|[[File:600-cell graph H4.svg|200px]]<BR>[30]
+|[[File:600-cell t0 p20.svg|200px]]<BR>[20]
+|[[File:600-cell t0 F4.svg|200px]]<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 t0 H3.svg|200px]]<BR>[10]
+|[[File:600-cell t0 A2.svg|200px]]<BR>[6]
+|[[File:600-cell t0.svg|200px]]<BR>[4]
+|}
+As a faceted 600-cell, replacing the [[tetrahedron|simplicial]] cells of the 600-cell with [[icosahedron|icosahedral]] [[pentagonal polytope]] cells, it could be seen as a four-dimensional analogue of the [[great dodecahedron]], which replaces the triangular faces of the icosahedron with pentagonal faces. Indeed, the icosahedral 120-cell is dual to the [[small stellated 120-cell]], which could be taken as a 4D analogue of the [[small stellated dodecahedron]], dual of the great dodecahedron.
 == See also ==
 * [[List of regular polytopes]]
-* [[Convex regular 4-polytope]] - Set of convex regular polychoron
+* [[Convex regular 4-polytope]]
 * [[Kepler-Poinsot solid]]s - 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].
+*[[Coxeter|H. S. M. Coxeter]], ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. {{ISBN|0-486-61480-8}}.
+* [[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)|x3o5o5/2o - fix}}
 == External links ==
-* [http://hometown.aol.com/hedrondude/regulars.html Regular polychora]
+* [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]
-* [http://www.mathematik.uni-regensburg.de/Goette/sterne Reguläre Polytope]
+* [https://web.archive.org/web/20061107052613/http://www.mathematik.uni-regensburg.de/Goette/sterne/ Reguläre Polytope]
-* [http://davidf.faricy.net/polyhedra/Star_Polychora.html The Regular Star Polychora]
+* [https://web.archive.org/web/20070704012333/http://davidf.faricy.net/polyhedra/Star_Polychora.html The Regular Star Polychora]
-{{polychora-stub}}
+{{Regular 4-polytopes}}
-[[Category:polychora]]
+[[Category:Regular 4-polytopes]]
+{{polychora-stub}}
-[[Category:4-dimensional geometry]]
-[[eo:Dudekedra 120-ĉelo]]
-[[fr:Hécatonicosachore icosaédral]]