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Cantellated 5-orthoplexes: Difference between revisions

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{| class=wikitable align=right width=540 style="margin-left:1em;"
|- align=center valign=top
|[[File:5-cube t4.svg|120px]]<BR>[[5-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|4|node}}
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|[[File:5-cube t234.svg|120px]]<BR>[[Cantitruncated 5-cube]]<BR>{{CDD|node|3|node|3|node_1|3|node_1|4|node_1}}
|-
!colspan=4|[[Orthogonal projection]]s in BCB<sub>5</sub> [[Coxeter plane]]
|}
In [[Five-dimensional space|five-dimensional]] [[geometry]], a '''cantellated 5-orthoplex''' is a convex [[uniform 5-polytope]], being a [[cantellation]] of the regular [[5-orthoplex]].
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== Cantellated 5-orthoplex==
 
{| class="wikitable" align="right" style="margin-left:10px" width="250290"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Cantellated 5-orthoplex'''
|-
|style="width:40%" bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
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|colspan=2| rr{3,3,3,4}<BR>rr{3,3,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
|colspan=2|{{CDD||node_1|3|node||3|node_1|3|node|4|node}}<BR>{{CDD||node_1|3|node||3|node_1|split1|nodes}}
|-
|bgcolor=#e7dcc3|4-faces
|82
|10 {{CDD|node|4|node|3|node_1|3|node}} [[File:Schlegel wireframe 24-cell.png|36px|link=24-cell]]<BR>40 {{CDD|node|4|node|3|node_1|2|node_1}} [[File:Triangular_antiprismatic_prism.png|36px|link=Octahedral prism]]<BR>32 {{CDD|node|3|node_1|3|node|3|node_1}} [[File:Schlegel_half-solid_cantellated_5-cell.png|36px|link=Cantellated 5-cell]]
|-
|bgcolor=#e7dcc3|Cells
|640
|80 {{CDD|node|4|node|3|node_1}} [[File:Uniform polyhedron-43-t2.png|25px|link=Octahedron]]<BR>160 {{CDD|node|3|node_1|3|node}} [[File:Uniform polyhedron-33-t1.png|25px|link=Octahedron]]<BR>320 {{CDD|node|3|node_1|2|node_1}} [[File:Triangular_prism_wedge.png|25px|link=Triangular prism]]<BR>80 {{CDD|node_1|3|node|3|node_1}} [[File:Uniform_polyhedron-33-t02.png|25px|link=Cuboctahedron]]
|-
|bgcolor=#e7dcc3|Faces
|1520
|640 {{CDD|node|3|node_1}} [[File:2-simplex_t0.svg|25px|link=Triangle]]<BR>320 {{CDD|node_1|3|node}} [[File:2-simplex_t0.svg|25px|link=Triangle]]<BR>480 {{CDD|node_1|2|node_1}} [[File:2-cube.svg|25px|link=Square]]<BR>80 {{CDD|node|3|node_1}} [[File:2-simplex_t0.svg|25px|link=Triangle]]
|-
|bgcolor=#e7dcc3|Edges
|1200
|960 {{CDD|node_1}}<BR>240 {{CDD|node_1}}
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|240
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|Square pyramidal prism [[File:Cantellated pentacross verf.png|40px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| BCB<sub>5</sub>, [4,3,3,3], order&nbsp;3840<BR>D<sub>5</sub>, [3<sup>2,1,1</sup>], order&nbsp;1920
|-
|bgcolor=#e7dcc3|Properties
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== Cantitruncated 5-orthoplex ==
 
{| class="wikitable" align="right" style="margin-left:10px" width="250290"
!bgcolor=#e7dcc3 colspan=23|Cantitruncated 5-orthoplex
|-
|style="width:100px" bgcolor=#e7dcc3|Type||colspan=2|[[uniform polyteron5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||colspan=2| tr{3,3,3,4}<BR>tr{3,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||colspan=2|{{CDD||node|4|node||3|node_1|3|node_1|3|node_1}}<BR>{{CDD|nodes|split2|node_1|3|node_1||3|node_1}}
|-
|bgcolor=#e7dcc3|4-faces||82
|style="width:30px"|82
|10 {{CDD|node|4|node|3|node_1|3|node_1}} [[File:Schlegel_half-solid_truncated_16-cell.png|36px|link=Truncated 16-cell]]<BR>40 {{CDD|node|4|node|3|node_1|2|node_1}} [[File:Triangular_antiprismatic_prism.png|36px|link=Octahedral prism]]<BR>32 {{CDD|node|3|node_1|3|node_1|3|node_1}} [[File:Schlegel_half-solid_cantitruncated_5-cell.png|36px|link=Cantitruncated 5-cell]]
|-
|bgcolor=#e7dcc3|Cells|
|640
|80 {{CDD|node|4|node|3|node_1}} [[File:Uniform polyhedron-43-t2.png|25px|link=Octahedron]]<BR>160 {{CDD|node|3|node_1|3|node_1}} [[File:Uniform polyhedron-33-t01.png|25px|link=Truncated tetrahedron]]<BR>320 {{CDD|node|3|node_1|2|node_1}} [[File:Triangular_prism_wedge.png|25px|link=Triangular prism]]<BR>80 {{CDD|node_1|3|node_1|3|node_1}} [[File:Uniform_polyhedron-33-t012.png|25px|link=Truncated octahedron]]
|-
|bgcolor=#e7dcc3|Faces|
|1520
|640 {{CDD|node|3|node_1}} [[File:2-simplex_t0.svg|25px|link=Triangle]]<BR>320 {{CDD|node_1|3|node_1}} [[File:2-simplex_t01.svg|25px|link=Hexagon]]<BR>480 {{CDD|node_1|2|node_1}} [[File:2-cube.svg|25px|link=Square]]<BR>80 {{CDD|node_1|3|node_1}} [[File:2-simplex_t01.svg|25px|link=Hexagon]]
|-
|bgcolor=#e7dcc3|Edges|
|1440
|960 {{CDD|node_1}}<BR>240 {{CDD|node_1}}<BR>240 {{CDD|node_1}}
|-
|bgcolor=#e7dcc3|Vertices||480
|colspan=2|480
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||colspan=2|[[Square pyramidal pyramid]] [[File:Canitruncated 5-orthoplex verf.png|40px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||BCcolspan=2|B<sub>5</sub>, [3,3,3,4], order&nbsp;3840<BR>D<sub>5</sub>, [3<sup>2,1,1</sup>], order&nbsp;1920
|-
|bgcolor=#e7dcc3|Properties||colspan=2|[[Convex polytope|convex]]
|}
=== Alternate names===
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== Related polytopes ==
 
These polytopes are from a set of 31 [[Uniform polyteron5-polytope#Uniform polyteron5-polytope|uniform polytera5-polytopes]] generated from the regular [[5-cube]] or [[5-orthoplex]].
 
{{Penteract family}}
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* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
** '''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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (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]
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