View source for Runcinated 120-cells

{| class=wikitable align=right
|+ Four runcinations
|- align=center valign=top
|[[File:120-cell t0 H3.svg|150px]]<BR>[[120-cell]]<BR>{{CDD|node_1|5|node|3|node|3|node}}
|[[File:120-cell t03 H3.png|150px]]<BR>Runcinated 120-cell<BR>(Expanded 120-cell)<BR>{{CDD|node_1|5|node|3|node|3|node_1}}
|[[File:120-cell t013 H3.png|150px]]<BR>Runcitruncated 120-cell<BR>{{CDD|node_1|5|node_1|3|node|3|node_1}}
|- align=center valign=top
|[[File:600-cell t0 H3.svg|150px]]<BR>[[600-cell]]<BR>{{CDD|node|5|node|3|node|3|node_1}}
|[[File:120-cell t023 H3.png|150px]]<BR>Runcitruncated 600-cell<BR>{{CDD|node_1|5|node|3|node_1|3|node_1}}
|[[File:120-cell t0123 H3.png|150px]]<BR>Omnitruncated 120-cell<BR>{{CDD|node_1|5|node_1|3|node_1|3|node_1}}
|-
!colspan=3|[[Orthogonal projection]]s in H<sub>3</sub> [[Coxeter plane]]
|}
In four-dimensional [[geometry]], a '''runcinated 120-cell''' (or ''runcinated 600-cell'') is a convex [[uniform 4-polytope]], being a [[runcination]] (a 3rd order truncation) of the regular [[120-cell]].

There are 4 degrees of runcinations of the 120-cell including with permutations truncations and cantellations.

The ''runcinated 120-cell'' can be seen as an [[Expansion (geometry)|expansion]] applied to a regular 4-polytope, the 120-cell or 600-cell.

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==Runcinated 120-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|'''Runcinated 120-cell'''
|-
|bgcolor=#e7dcc3|Type||[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|Uniform index||38
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||2640 total:<BR>120 [[Dodecahedron|5.5.5]] [[Image:Dodecahedron.png|20px]]<BR>720 [[pentagonal prism|4.4.5]] [[Image:Pentagonal prism.png|20px]]<BR>1200 [[triangular prism|4.4.3]] [[Image:triangular prism.png|20px]]<BR>600 [[tetrahedron|3.3.3]] [[Image:tetrahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces||7440:<BR>2400{3}+3600{4}+<BR>1440{5}
|-
|bgcolor=#e7dcc3|Edges||7200
|-
|bgcolor=#e7dcc3|Vertices||2400
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Runcinated 120-cell verf.png|80px]]<BR>Equilateral-triangular antipodium
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,3</sub>{5,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]||H<sub>4</sub>, [3,3,5], order 14400
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
[[File:Small disprismatohexacosihecatonicosachoron net.png|thumb|right|[[Net (polyhedron)|Net]]]]
The '''runcinated 120-cell''' or '''small disprismatohexacosihecatonicosachoron''' is a [[uniform 4-polytope]]. It has 2640 cells: 120 [[Dodecahedron|dodecahedra]], 720 [[pentagonal prism]]s, 1200 [[triangular prism]]s, and 600 [[tetrahedron|tetrahedra]]. Its [[vertex figure]] is a nonuniform triangular [[antiprism]] (equilateral-triangular antipodium): its bases represent a dodecahedron and a tetrahedron, and its flanks represent three triangular prisms and three pentagonal prisms.

=== Alternate names ===
* Runcinated 120-cell / Runcinated 600-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]])
** Runcinated hecatonicosachoron / Runcinated dodecacontachoron / Runcinated hexacosichoron / Runcinated polydodecahedron / Runcinated polytetrahedron
* Small diprismatohexacosihecatonicosachoron (acronym: sidpixhi) (George Olshevsky, Jonathan Bowers)<ref>Klitizing, (x3o3o5x - sidpixhi)</ref>

=== Images===
{| class=wikitable width=400
|+ [[Schlegel diagram]] (Only tetrahedral cells shown)
|[[Image:Runcinated 120-cell.png|400px]]
|}

{| class=wikitable width=480
|+ Polyhedral rings
|[[File:Runcinated_120-cell-5-fold-ring-cells.png|160px]]<BR>Cells on 5-fold axis
|[[File:Runcinated_120-cell-3-fold-ring-cells.png|160px]]<BR>Cells on 3-fold axis
|[[File:Runcinated_120-cell-2-fold-ring-cells.png|160px]]<BR>Cells on 2-fold axis
|}
{| class=wikitable
|+ [[Orthogonal projection]]s in [[Coxeter plane]]s
|- valign=top align=center
|[[Image:120-cell t03 H3.png|160px]]<BR>H3
|[[Image:120-cell t03 B3.png|160px]]<BR>A2/B3
|[[Image:120-cell t03 A3.png|160px]]<BR>A3/B2
|}

{{-}}

== Runcitruncated 120-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|'''Runcitruncated 120-cell'''
|-
|bgcolor=#e7dcc3|Type||[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|Uniform index||43
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node_1|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||2640 total:<BR>120 [[Truncated dodecahedron|(3.10.10)]] [[Image:Truncated dodecahedron.png|20px]]<BR>
720 [[Decagonal prism|(4.4.10)]] [[Image:Decagonal prism.png|20px]]<BR>1200 [[Triangular prism|(3.4.4)]] [[Image:Triangular prism.png|20px]]<BR>600 [[Cuboctahedron|(3.4.3.4)]] [[Image:Cuboctahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces||13440:<BR>4800{3}+7200{4}+<BR>1440{10}
|-
|bgcolor=#e7dcc3|Edges||18000
|-
|bgcolor=#e7dcc3|Vertices||7200
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Runcitruncated 120-cell verf.png|80px]]<BR>Irregular rectangular pyramid
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,3</sub>{5,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]||H<sub>4</sub>, [3,3,5], order 14400
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
[[File:Prismatorhombated hexacosichoron net.png|thumb|right|[[Net (polyhedron)|Net]]]]
The '''runcitruncated 120-cell''' or '''prismatorhombated hexacosichoron''' is a [[uniform 4-polytope]]. It contains 2640 cells: 120 [[Truncated dodecahedron|truncated dodecahedra]], 720 [[decagonal prism]]s, 1200 [[triangular prism]]s, and 600 [[Cuboctahedron|cuboctahedra]]. Its [[vertex figure]] is an irregular rectangular pyramid, with one truncated dodecahedron, two decagonal prisms, one triangular prism, and one cuboctahedron.

=== Alternate names ===
* Runcicantellated 600-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]])
* Prismatorhombated hexacosichoron (Acronym: prix) (George Olshevsky, Jonathan Bowers)<ref>Klitizing, (x3o3x5x - prix)</ref>

=== Images===
{| class=wikitable
|+ [[Schlegel diagram]] (Only triangular prisms shown)
|colspan=3 align=center|[[Image:Runcitruncated 120-cell.png|400px]]
|}

{| class=wikitable
|+ [[Orthogonal projection]]s in [[Coxeter plane]]s
|-
|valign=top align=center|[[Image:120-cell t013 H3.png|160px]]<BR>H3
|valign=top align=center|[[Image:120-cell t013 B3.png|160px]]<BR>A2/B3
|valign=top align=center|[[Image:120-cell t013 A3.png|160px]]<BR>A3/B2
|}

{{-}}

== Runcitruncated 600-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|'''Runcitruncated 600-cell'''
|-
|bgcolor=#e7dcc3|Type||[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|Uniform index||44
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||2640 total:<BR>120 [[rhombicosidodecahedron|3.4.5.4]] [[Image:Small rhombicosidodecahedron.png|20px]]<BR>720 [[Pentagonal prism|4.4.5]] [[Image:Pentagonal prism.png|20px]]<BR>1200 [[Hexagonal prism|4.4.6]] [[Image:Hexagonal prism.png|20px]]<BR>600 [[Truncated tetrahedron|3.6.6]] [[Image:Truncated tetrahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces||13440:<BR>2400{3}+7200{4}+<BR>1440{5}+2400{6}
|-
|bgcolor=#e7dcc3|Edges||18000
|-
|bgcolor=#e7dcc3|Vertices||7200
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Runcitruncated 600-cell verf.png|80px]]<BR>Trapezoidal pyramid
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,3</sub>{3,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]||H<sub>4</sub>, [3,3,5], order 14400
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
[[File:Prismatorhombated hecatonicosachoron net.png|thumb|right|[[Net (polyhedron)|Net]]]]
The '''runcitruncated 600-cell''' or '''prismatorhombated hecatonicosachoron''' is a [[uniform 4-polytope]]. It is composed of 2640 [[cell (mathematics)|cells]]: 120 [[rhombicosidodecahedron]], 600 [[Truncated tetrahedron|truncated tetrahedra]], 720 [[pentagonal prism]]s, and 1200 [[hexagonal prism]]s. It has 7200 vertices, 18000 edges, and 13440 faces (2400 triangles, 7200 squares, and 2400 hexagons).

=== Alternate names ===
* Runcicantellated 120-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]])
* Prismatorhombated hecatonicosachoron (Acronym: prahi) (George Olshevsky, Jonathan Bowers)<ref>Klitizing, (x3x3o5x - prahi)</ref>

=== Images===
{| class=wikitable
|+ [[Schlegel diagram]]
|[[Image:Runcitruncated 600-cell.png|400px]]
|}

{| class=wikitable
|+ [[Orthogonal projection]]s in [[Coxeter plane]]s
|- valign=top align=center
|[[Image:120-cell t023 H3.png|160px]]<BR>H3
|[[Image:120-cell t023 B3.png|160px]]<BR>A2/B3
|[[Image:120-cell t023 A3.png|160px]]<BR>A3/B2
|}

{{-}}

==Omnitruncated 120-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|'''Omnitruncated 120-cell'''
|-
|bgcolor=#e7dcc3|Type||[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|Uniform index||46
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node_1|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||2640 total:<BR>120 [[Truncated icosidodecahedron|4.6.10]] [[Image:Great rhombicosidodecahedron.png|20px]]<BR>720 [[decagonal prism|4.4.10]] [[Image:Decagonal prism.png|20px]]<BR>1200 [[Hexagonal prism|4.4.6]] [[Image:Hexagonal prism.png|20px]]<BR>600 [[Truncated octahedron|4.6.6]] [[Image:Truncated octahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces||17040 total:<BR>10800 [[square (geometry)|{4}]], 4800 [[hexagon|{6}]]<BR>1440 [[decagon|{10}]]
|-
|bgcolor=#e7dcc3|Edges||28800
|-
|bgcolor=#e7dcc3|Vertices||14400
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Omnitruncated 120-cell verf.png|80px]]<BR>Chiral scalene tetrahedron
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,2,3</sub>{3,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]||H<sub>4</sub>, [3,3,5], order 14400
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''omnitruncated 120-cell''' or '''great disprismatohexacosihecatonicosachoron''' is a convex [[uniform 4-polytope]], composed of 2640 [[cell (mathematics)|cells]]: 120 [[truncated icosidodecahedron|truncated icosidodecahedra]], 600 [[truncated octahedron|truncated octahedra]], 720 [[decagonal prism]]s, and 1200 [[hexagonal prism]]s. It has 14400 vertices, 28800 edges, and 17040 faces (10800 squares, 4800 hexagons, and 1440 decagons). It is the largest nonprismatic convex [[uniform 4-polytope]].

The vertices and edges form the [[Cayley graph]] of the [[Coxeter group]] H<sub>4</sub>.

=== Alternate names===
* Omnitruncated 120-cell / Omnitruncated 600-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]])
* Omnitruncated hecatonicosachoron / Omnitruncated hexacosichoron / Omnitruncated polydodecahedron / Omnitruncated polytetrahedron
* Great diprismatohexacosihecatonicosachoron (Acronym gidpixhi) (George Olshevsky, Jonathan Bowers)<ref>Klitizing, (x3x3x5x - gidpixhi)</ref>

=== Images ===
{| class=wikitable width=480
|[[Image:Omnitruncated 120-cell wireframe.png|240px]]
|valign=top align=center|[[Image:Stereographic omnitruncated 120-cell.png|240px]]
|-
![[Schlegel diagram]] (centered on [[truncated icosidodecahedron]])<BR>(Orthogonal view, centered on [[decagonal prism]] cell.)
![[Stereographic projection]]<BR>(centered on [[truncated icosidodecahedron]])
|}

{| class=wikitable width=480
|+ [[Orthogonal projection]]s in [[Coxeter plane]]s
|- valign=top align=center
|[[Image:120-cell t0123 H3.png|160px]]<BR>H3
|[[Image:120-cell t0123 B3.png|160px]]<BR>A2/B3
|[[Image:120-cell t0123 A3.png|160px]]<BR>A3/B2
|}
{| class=wikitable width=480
|+ Polyhedral rings
|[[File:Omnitruncated_120-cell-5-fold-ring-cells.png|160px]]<BR>Cells on 5-fold axis
|[[File:Omnitruncated_120-cell-3-fold-ring-cells.png|160px]]<BR>Cells on 3-fold axis
|[[File:Omnitruncated_120-cell-2-fold-ring-cells.png|160px]]<BR>Cells on 2-fold axis
|}

{| class=wikitable
|+ [[Net (polyhedron)|Net]]
|[[Image:Omnitruncated 120-cell net.png|200px]]<BR> Omnitruncated 120-cell
|[[File:Dual_gidpixhi_net.png|200px]]<BR>Dual to omnitruncated 120-cell
|}

===Models===
The first complete physical model of a 3D projection of the omnitruncated 120-cell was built by a team led by Daniel Duddy and David Richter on August 9, 2006 using the [[Zome]] system in the London Knowledge Lab for the 2006 [http://www.bridgesmathart.org/ Bridges Conference].<ref>[http://homepages.wmich.edu/~drichter/bridgeszome2006.htm Photos of Zome model of omnitruncated 120/600-cell]</ref>

=== Full snub 120-cell ===
[[File:Snub 120-cell verf.png|80px|thumb|Vertex figure for the '''omnisnub 120-cell''']]
The '''full snub 120-cell''' or '''omnisnub 120-cell''', defined as an [[Alternation (geometry)|alternation]] of the omnitruncated 120-cell, can not be made uniform, but it can be given Coxeter diagram {{CDD|node_h|5|node_h|3|node_h|3|node_h}}, and [[Coxeter notation|symmetry]] [5,3,3]<sup>+</sup>, and constructed from 1200 [[octahedron]]s, 600 [[icosahedron]]s, 720 [[pentagonal antiprism]]s, 120 [[snub dodecahedron]]s, and 7200 [[tetrahedron]]s filling the gaps at the deleted vertices. It has 9840 cells, 35040 faces, 32400 edges, and 7200 vertices.<ref>{{Cite web|url=http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm|title = S3s3s5s}}</ref>

== Related polytopes==
These polytopes are a part of a set of 15 [[uniform 4-polytope]]s with H<sub>4</sub> symmetry:

{{H4_family}}

== Notes==
{{reflist}}

== References ==
* ''[http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Kaleidoscopes: Selected Writings of H.S.M. Coxeter]'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}}
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[John Horton Conway|J.H. Conway]] and [[Michael Guy (computer scientist)|M.J.T. Guy]]: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
*[http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation  [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] [http://www.polytope.de/nr55.html m55] [http://www.polytope.de/nr62.html m62] [http://www.polytope.de/nr60.html m60] [http://www.polytope.de/nr64.html m64]
* {{PolyCell | urlname = section4.html| title = Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 38, 44, 47}}
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} x3o3o5x - sidpixhi, x3o3x5x - prix, x3x3o5x - prahi, x3x3x5x - gidpixhi

== External links==
* H4 uniform polytopes with coordinates: [http://www.qfbox.info/4d/runci120cell t03{5,3,3}] [http://www.qfbox.info/4d/runcitrunc600cell t013{3,3,5}] [http://www.qfbox.info/4d/runcitrunc120cell t013{5,3,3}] [http://www.qfbox.info/4d/omni120cell t0123{5,3,3}] 

{{Polytopes}}

{{DEFAULTSORT:Runcinated 120-Cell}}
[[Category:Uniform 4-polytopes]]