View source for Stericated 5-cubes

{| class=wikitable align=right width=480 style="margin-left:1em;"
|- align=center valign=top
|[[File:5-cube t0.svg|160px]]<BR><small>[[5-cube]]</small><BR>{{CDD|node_1|4|node|3|node|3|node|3|node}}
|[[File:5-cube t04.svg|160px]]<BR><small>Stericated 5-cube</small><BR>{{CDD|node_1|4|node|3|node|3|node|3|node_1}}
|[[File:5-cube t014.svg|160px]]<BR><small>Steritruncated 5-cube</small><BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node_1}}
|- align=center valign=top
|[[File:5-cube t024.svg|160px]]<BR><small>Stericantellated 5-cube</small><BR>{{CDD|node_1|4|node|3|node_1|3|node|3|node_1}}
|[[File:5-cube t034.svg|160px]]<BR><small>[[Steritruncated 5-orthoplex]]</small><BR>{{CDD|node_1|4|node|3|node|3|node_1|3|node_1}}
|[[File:5-cube t0124.svg|160px]]<BR><small>Stericantitruncated 5-cube</small><BR>{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1}}
|- align=center valign=top
|[[File:5-cube t0134.svg|160px]]<BR><small>Steriruncitruncated 5-cube</small><BR>{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1}}
|[[File:5-cube t0234.svg|160px]]<BR><small>[[Stericantitruncated 5-orthoplex]]</small><BR>{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1}}
|[[File:5-cube t01234.svg|160px]]<BR>[[Omnitruncated 5-cube]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1}}
|-
!colspan=3|[[Orthogonal projection]]s in B<sub>5</sub> [[Coxeter plane]]
|}
In [[Five-dimensional space|five-dimensional]] [[geometry]], a '''stericated 5-cube''' is a convex [[uniform 5-polytope]] with fourth-order [[Truncation (geometry)|truncations]] ([[sterication]]) of the regular [[5-cube]].

There are eight degrees of sterication for the 5-cube, including permutations of [[runcination]], [[cantellation]], and [[Truncation (geometry)|truncation]]. The simple stericated 5-cube is also called an '''expanded 5-cube''', with the first and last nodes ringed, for being [[constructible polygon|constructible]] by an [[Expansion (geometry)|expansion]] operation applied to the regular 5-cube. The highest form, the '''sterirunci{{wbr}}cantitruncated{{wbr}} 5-cube''', is more simply called an [[#Omnitruncated 5-cube|omnitruncated 5-cube]] with all of the nodes ringed.
{{clear}}

== Stericated 5-cube ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Stericated 5-cube'''
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| 2r2r{4,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD||node_1|4|node||3|node|3|node|3|node_1}}<BR>{{CDD|node|split1|nodes|3a4b|nodes_11}}
|-
|bgcolor=#e7dcc3|4-faces
|242
|-
|bgcolor=#e7dcc3|Cells
|800
|-
|bgcolor=#e7dcc3|Faces
|1040
|-
|bgcolor=#e7dcc3|Edges
|640
|-
|bgcolor=#e7dcc3|Vertices
|160
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Stericated penteract verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| B<sub>5</sub> [4,3,3,3]
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|}

=== Alternate names ===
* Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
* Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
* Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)<ref>Klitzing, (x3o3o3o4x - scant)</ref>

=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of a ''stericated 5-cube'' having edge length&nbsp;2 are all permutations of:

:<math>\left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2})\right)</math>

=== Images ===
The stericated 5-cube is constructed by a [[sterication]] operation applied to the 5-cube.

=== Dissections ===
The stericated 5-cube can be dissected into two [[Cupola (geometry)|tesseractic cupolae]] and a [[Runcinated tesseracts|runcinated tesseract]] between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into two [[Cubic cupola|cubic cupolae]] and a central [[rhombicuboctahedral prism]] between them, and also the 3D rhombicuboctahedron being dissected into two [[Square cupola|square cupolae]] with a central [[octagonal prism]] between them.{{5-cube Coxeter plane graphs|t04|150}}

==Steritruncated 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steritruncated 5-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,4</sub>{4,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|node_1|3|node|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|4-faces||242
|-
|bgcolor=#e7dcc3|Cells||1600
|-
|bgcolor=#e7dcc3|Faces||2960
|-
|bgcolor=#e7dcc3|Edges||2240
|-
|bgcolor=#e7dcc3|Vertices||640
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Steritruncated 5-cube verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>5</sub>, [3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

===Alternate names===
* Steritruncated penteract
* Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)<ref>Klitzing, (x3o3o3x4x - capt)</ref>

===Construction and coordinates===

The [[Cartesian coordinate]]s of the vertices of a ''steritruncated 5-cube'' having edge length&nbsp;2 are all permutations of:

:<math>\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)</math>

=== Images ===

{{5-cube Coxeter plane graphs|t014|150}}

==Stericantellated 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Stericantellated 5-cube'''
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| t<sub>0,2,4</sub>{4,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD||node_1|4|node|3|node_1|3|node|3|node_1}}<BR>{{CDD|node_1|split1|nodes|3a4b|nodes_11}}
|-
|bgcolor=#e7dcc3|4-faces||242
|-
|bgcolor=#e7dcc3|Cells||2080
|-
|bgcolor=#e7dcc3|Faces||4720
|-
|bgcolor=#e7dcc3|Edges||3840
|-
|bgcolor=#e7dcc3|Vertices||960
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Stericantellated 5-cube verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| B<sub>5</sub> [4,3,3,3]
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|}

=== Alternate names ===
* Stericantellated penteract
* Stericantellated 5-orthoplex, stericantellated pentacross
* Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)<ref>Klitzing, (x3o3x3o4x - carnit)</ref>

=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of a ''stericantellated 5-cube'' having edge length&nbsp;2 are all permutations of:

:<math>\left(\pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)</math>

=== Images ===

{{5-cube Coxeter plane graphs|t024|150}}

==Stericantitruncated 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Stericantitruncated 5-cube'''
|-
|bgcolor=#e7dcc3|Type
|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|t<sub>0,1,2,4</sub>{4,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|4-faces||242
|-
|bgcolor=#e7dcc3|Cells||2400
|-
|bgcolor=#e7dcc3|Faces||6000
|-
|bgcolor=#e7dcc3|Edges||5760
|-
|bgcolor=#e7dcc3|Vertices||1920
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Stericanitruncated 5-cube verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| B<sub>5</sub> [4,3,3,3]
|-
|bgcolor=#e7dcc3|Properties
|[[Convex polytope|convex]], [[isogonal figure|isogonal]]
|}

=== Alternate names ===
* Stericantitruncated penteract
* Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
* Celligreatorhombated penteract (cogrin) (Jonathan Bowers)<ref>Klitzing, (x3o3x3x4x - cogrin)</ref>

=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of a stericantitruncated 5-cube having an edge length of&nbsp;2 are given by all permutations of coordinates and sign of:

:<math>\left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right)</math>

=== Images ===
{{5-cube Coxeter plane graphs|t013|150}}

==Steriruncitruncated 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Steriruncitruncated 5-cube'''
|-
|bgcolor=#e7dcc3|Type
|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|2t2r{4,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1}}<BR>{{CDD|node|split1|nodes_11|3a4b|nodes_11}}
|-
|bgcolor=#e7dcc3|4-faces||242
|-
|bgcolor=#e7dcc3|Cells||2160
|-
|bgcolor=#e7dcc3|Faces||5760
|-
|bgcolor=#e7dcc3|Edges||5760
|-
|bgcolor=#e7dcc3|Vertices||1920
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Steriruncitruncated 5-cube verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| B<sub>5</sub> [4,3,3,3]
|-
|bgcolor=#e7dcc3|Properties
|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
|}

=== Alternate names ===
* Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
* Celliprismatotruncated penteractitriacontiditeron  (captint) (Jonathan Bowers)<ref>Klitzing, (x3x3o3x4x - captint)</ref>

=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of a steriruncitruncated penteract having an edge length of&nbsp;2 are given by all permutations of coordinates and sign of:

:<math>\left(1,\ 1+\sqrt{2},\ 1+1\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right)</math>

=== Images ===
{{5-cube Coxeter plane graphs|t0134|150}}

==Steritruncated 5-orthoplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steritruncated 5-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,4</sub>{3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|node|3|node|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|4-faces||242
|-
|bgcolor=#e7dcc3|Cells||1520
|-
|bgcolor=#e7dcc3|Faces||2880
|-
|bgcolor=#e7dcc3|Edges||2240
|-
|bgcolor=#e7dcc3|Vertices||640
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Steritruncated 5-orthoplex verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||B<sub>5</sub>, [3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

===Alternate names===
* Steritruncated pentacross
* Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)<ref>Klitzing, (x3x3o3o4x - cappin)</ref>

=== Coordinates ===
[[Cartesian coordinates]] for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all [[permutation]]s of
:<math>\left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2})\right)</math>

=== Images ===
{{5-cube Coxeter plane graphs|t034|150}}

==Stericantitruncated 5-orthoplex==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Stericantitruncated 5-orthoplex'''
|-
|bgcolor=#e7dcc3|Type
|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|t<sub>0,2,3,4</sub>{4,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|4-faces||242
|-
|bgcolor=#e7dcc3|Cells||2320
|-
|bgcolor=#e7dcc3|Faces||5920
|-
|bgcolor=#e7dcc3|Edges||5760
|-
|bgcolor=#e7dcc3|Vertices||1920
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Stericanitruncated 5-orthoplex verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| B<sub>5</sub> [4,3,3,3]
|-
|bgcolor=#e7dcc3|Properties
|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
|}

=== Alternate names ===
* Stericantitruncated pentacross
* Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)<ref>Klitzing, (x3x3x3o4x - cogart)</ref>

=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of a stericantitruncated 5-orthoplex having an edge length of&nbsp;2 are given by all permutations of coordinates and sign of:

:<math>\left(1,\ 1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right)</math>

=== Images ===
{{5-cube Coxeter plane graphs|t0234|150}}

==Omnitruncated 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Omnitruncated 5-cube'''
|-
|bgcolor=#e7dcc3|Type
|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|tr2r{4,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
|{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1}}<BR>{{CDD|node_1|split1|nodes_11|3a4b|nodes_11}}
|-
|bgcolor=#e7dcc3|4-faces||242
|-
|bgcolor=#e7dcc3|Cells||2640
|-
|bgcolor=#e7dcc3|Faces||8160
|-
|bgcolor=#e7dcc3|Edges||9600
|-
|bgcolor=#e7dcc3|Vertices||3840
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Omnitruncated 5-cube verf.png|80px]]<BR>irr. {3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| B<sub>5</sub> [4,3,3,3]
|-
|bgcolor=#e7dcc3|Properties
|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
|}

=== Alternate names ===
* Steriruncicantitruncated 5-cube (Full expansion of [[omnitruncation]] for 5-polytopes by Johnson)
* Omnitruncated penteract
* Omnitruncated triacontiditeron / omnitruncated pentacross
* Great cellated penteractitriacontiditeron (Jonathan Bowers)<ref>Klitzing, (x3x3x3x4x - gacnet)</ref>

=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of an omnitruncated 5-cube having an edge length of&nbsp;2 are given by all permutations of coordinates and sign of:

:<math>\left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2},\ 1+4\sqrt{2}\right)</math>

=== Images ===
{{5-cube Coxeter plane graphs|t01234|150}}

=== Full snub 5-cube ===

The '''full snub 5-cube''' or '''omnisnub 5-cube''', defined as an [[Alternation (geometry)|alternation]] of the omnitruncated 5-cube is not uniform, but it can be given Coxeter diagram {{CDD|node_h|4|node_h|3|node_h|3|node_h|3|node_h}} and [[Coxeter notation|symmetry]] [4,3,3,3]<sup>+</sup>, and constructed from 10 [[Runcinated tesseracts#Full snub tesseract|snub tesseract]]s, 32 [[Runcinated 5-cell#Full snub 5-cell|snub 5-cell]]s, 40 [[Truncated cuboctahedral prism#Related polytopes|snub cubic antiprism]]s, 80 [[Truncated octahedral prism#Related polytopes|snub tetrahedral antiprism]]s, 80 [[6-8 duoprism#Related polytopes|3-4 duoantiprism]]s, and 1920 irregular [[5-cell]]s filling the gaps at the deleted vertices.

== Related polytopes ==
This polytope is one of 31 [[Uniform_5-polytope#Uniform_5-polytope|uniform 5-polytopes]] generated from the regular [[5-cube]] or [[5-orthoplex]].

{{Penteract family}}

==Notes==
{{reflist}}

== References ==
* [[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''', editied 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]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 
* {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} x3o3o3o4x - scan, x3o3o3x4x - capt, x3o3x3o4x - carnit, x3o3x3x4x - cogrin, x3x3o3x4x - captint, x3x3x3x4x - gacnet, x3x3x3o4x - cogart

== External links ==
* {{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:5-polytopes]]