|
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| Line 1: |
Line 1: |
| {| class=wikitable align=right width=520
| |
| |- align=center valign=top
| |
| |[[File:5-simplex t0.svg|120px]] [[File:5-simplex t0 A4.svg|120px]]<BR><small>[[5-simplex]]</small><BR>{{CDD|node_1|3|node|3|node|3|node|3|node}}
| |
| |[[File:5-simplex t04.svg|120px]] [[File:5-simplex t04 A4.svg|120px]]<BR><small>'''Stericated 5-simplex'''</small><BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1}}
| |
| |- align=center valign=top
| |
| |[[File:5-simplex t014.svg|120px]] [[File:5-simplex t014 A4.svg|120px]]<BR><small>'''Steritruncated 5-simplex'''</small><BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node_1}}
| |
| |[[File:5-simplex t024.svg|120px]] [[File:5-simplex t024 A4.svg|120px]]<BR><small>'''Stericantellated 5-simplex'''</small><BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node_1}}
| |
| |- align=center valign=top
| |
| |[[File:5-simplex t0124.svg|120px]] [[File:5-simplex t0124 A4.svg|120px]]<BR><small>'''Stericantitruncated 5-simplex'''</small><BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node_1}}
| |
| |[[File:5-simplex t0134.svg|120px]] [[File:5-simplex t0134 A4.svg|120px]]<BR><small>'''Steriruncitruncated 5-simplex'''</small><BR>{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1}}
| |
| |- align=center valign=top
| |
| |colspan=2|[[File:5-simplex t01234.svg|180px]] [[File:5-simplex t01234 A4.svg|180px]]<BR><small>'''Steriruncicantitruncated 5-simplex'''</small><BR>(Omnitruncated 5-simplex)<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
| |
| |-
| |
| !colspan=4|[[Orthogonal projection]]s in A<sub>5</sub> and A<sub>4</sub> [[Coxeter plane]]s
| |
| |}
| |
|
| |
|
| In five-dimensional [[geometry]], a '''stericated 5-simplex''' is a convex [[uniform 5-polytope]] with fourth-order [[Truncation (geometry)|truncations]] ([[sterication]]) of the regular [[5-simplex]].
| |
|
| |
|
| There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an '''expanded 5-simplex''', with the first and last nodes ringed, for being constructible by an [[Expansion (geometry)|expansion]] operation applied to the regular 5-simplex. The highest form, the ''steriruncicantitruncated 5-simplex'' is more simply called an [[#Omnitruncated 5-simplex|omnitruncated 5-simplex]] with all of the nodes ringed.
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| | |
| == Stericated 5-simplex ==
| |
| {| class="wikitable" align="right" style="margin-left:10px" width="280"
| |
| |-
| |
| |bgcolor=#e7dcc3 align=center colspan=3|'''Stericated 5-simplex'''
| |
| |-
| |
| |bgcolor=#e7dcc3|Type
| |
| |colspan=2|[[Uniform 5-polytope]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Schläfli symbol]]
| |
| |colspan=2|2r2r{3,3,3,3}
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
| |
| |colspan=2|{{CDD||node_1|3|node||3|node||3|node||3|node_1}}<BR>or {{CDD|node|split1|nodes|3ab|nodes_11}}
| |
| |-
| |
| |bgcolor=#e7dcc3|4-faces
| |
| |62
| |
| |6+6 [[Pentachoron|{3,3,3}]][[Image:Schlegel wireframe 5-cell.png|25px]]<BR>15+15 [[Tetrahedral prism|{}×{3,3}]][[Image:Tetrahedral prism.png|25px]]<BR>20 [[Duoprism|{3}×{3}]][[Image:3-3 duoprism.png|25px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Cells
| |
| |180
| |
| |60 [[Tetrahedron|{3,3}]][[Image:Tetrahedron.png|25px]]<BR>120 [[Triangular prism|{}×{3}]][[Image:Triangular prism.png|25px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Faces
| |
| |210
| |
| |120 [[Triangle|{3}]]<BR>90 [[Square (geometry)|{4}]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Edges
| |
| |colspan=2|120
| |
| |-
| |
| |bgcolor=#e7dcc3|Vertices
| |
| |colspan=2|30
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Vertex figure]]
| |
| |colspan=2|[[File:Stericated hexateron verf.png|80px]]<BR>[[16-cell|Tetrahedral antiprism]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter group]]
| |
| |colspan=2|A<sub>5</sub>×2, [<span/>[3,3,3,3]], order 1440
| |
| |-
| |
| |bgcolor=#e7dcc3|Properties
| |
| |colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]], [[Isotoxal figure|isotoxal]]
| |
| |}
| |
| | |
| A '''stericated 5-simplex''' can be constructed by an [[Expansion (geometry)|expansion]] operation applied to the regular [[5-simplex]], and thus is also sometimes called an '''expanded 5-simplex'''. It has 30 [[Vertex (geometry)|vertices]], 120 [[Edge (geometry)|edges]], 210 [[Face (geometry)|faces]] (120 [[triangle]]s and 90 [[Square (geometry)|squares]]), 180 [[Cell (geometry)|cells]] (60 [[Tetrahedron|tetrahedra]] and 120 [[triangular prism]]s) and 62 [[4-face]]s (12 [[5-cell]]s, 30 [[tetrahedral prism]]s and 20 [[Duoprism|3-3 duoprisms]]).
| |
| | |
| === Alternate names ===
| |
| * Expanded 5-simplex
| |
| * Stericated hexateron
| |
| * Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)<ref>Klitizing, (x3o3o3o3x - scad)</ref>
| |
| | |
| ===Cross-sections===
| |
| | |
| The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a [[runcinated pentachoron]]. This cross-section divides the stericated hexateron into two [[Cupola (geometry)#Hypercupolas|pentachoral hypercupolas]] consisting of 6 [[Pentachoron|pentachora]], 15 [[tetrahedral prism]]s and 10 [[Duoprism|3-3 duoprisms]] each.
| |
| | |
| === Coordinates ===
| |
| | |
| The vertices of the ''stericated 5-simplex'' can be constructed on a [[hyperplane]] in 6-space as permutations of (0,1,1,1,1,2). This represents the positive [[orthant]] [[Facet (geometry)|facet]] of the [[stericated 6-orthoplex]].
| |
| | |
| A second construction in 6-space, from the center of a [[rectified 6-orthoplex]] is given by coordinate permutations of:
| |
| : (1,-1,0,0,0,0)
| |
| | |
| The [[Cartesian coordinates]] in 5-space for the normalized vertices of an origin-centered '''stericated hexateron''' are:
| |
| | |
| :<math>\left(\pm1,\ 0,\ 0,\ 0,\ 0\right)</math>
| |
| :<math>\left(0,\ \pm1,\ 0,\ 0,\ 0\right)</math>
| |
| :<math>\left(0,\ 0,\ \pm1,\ 0,\ 0\right)</math>
| |
| :<math>\left(\pm1/2,\ 0,\ \pm1/2,\ -\sqrt{1/8},\ -\sqrt{3/8}\right)</math>
| |
| :<math>\left(\pm1/2,\ 0,\ \pm1/2,\ \sqrt{1/8},\ \sqrt{3/8}\right)</math>
| |
| :<math>\left( 0,\ \pm1/2,\ \pm1/2,\ -\sqrt{1/8},\ \sqrt{3/8}\right)</math>
| |
| :<math>\left( 0,\ \pm1/2,\ \pm1/2,\ \sqrt{1/8},\ -\sqrt{3/8}\right)</math>
| |
| :<math>\left(\pm1/2,\ \pm1/2,\ 0,\ \pm\sqrt{1/2},\ 0\right)</math>
| |
| | |
| === Root system ===
| |
| Its 30 vertices represent the root vectors of the [[simple Lie group]] A<sub>5</sub>. It is also the [[vertex figure]] of the [[5-simplex honeycomb]].
| |
| | |
| === Images ===
| |
| | |
| {{5-simplex2 Coxeter plane graphs|t04|100}}
| |
| | |
| {| class=wikitable
| |
| |- align=center
| |
| |[[File:Stericated hexateron ortho.svg|160px]]<BR>orthogonal projection with [6] symmetry
| |
| |}
| |
| | |
| ==Steritruncated 5-simplex==
| |
| {| class="wikitable" align="right" style="margin-left:10px" width="280"
| |
| |-
| |
| |bgcolor=#e7dcc3 align=center colspan=3|'''Steritruncated 5-simplex'''
| |
| |-
| |
| |bgcolor=#e7dcc3|Type
| |
| |colspan=2|[[Uniform 5-polytope]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Schläfli symbol]]
| |
| |colspan=2|t<sub>0,2,3</sub>{3,3,3,3}
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
| |
| |colspan=2|{{CDD|node_1|3|node_1||3|node|3|node|3|node_1}}
| |
| |-
| |
| |bgcolor=#e7dcc3|4-faces
| |
| |62
| |
| |6 [[Truncated 5-cell|t{3,3,3}]]<BR>15 {}×[[Truncated tetrahedron|t{3,3}]]<BR>20 [[Duoprism|{3}×{6}]]<BR>15 {}×[[Tetrahedron|{3,3}]]<BR>6 [[Runcinated 5-cell|rr{3,3,3}]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Cells
| |
| |330
| |
| |
| |
| |-
| |
| |bgcolor=#e7dcc3|Faces
| |
| |570
| |
| |
| |
| |-
| |
| |bgcolor=#e7dcc3|Edges
| |
| |colspan=2|420
| |
| |-
| |
| |bgcolor=#e7dcc3|Vertices
| |
| |colspan=2|120
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Vertex figure]]
| |
| |colspan=2|[[File:Steritruncated 5-simplex verf.png|100px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter group]]
| |
| |colspan=2|A<sub>5</sub> [3,3,3,3], order 720
| |
| |-
| |
| |bgcolor=#e7dcc3|Properties
| |
| |colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
| |
| |}
| |
| | |
| === Alternate names ===
| |
| * Steritruncated hexateron
| |
| * Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)<ref>Klitizing, (x3x3o3o3x - cappix)</ref>
| |
| | |
| === Coordinates ===
| |
| The coordinates can be made in 6-space, as 180 permutations of:
| |
| : (0,1,1,1,2,3)
| |
| | |
| This construction exists as one of 64 [[orthant]] [[Facet (geometry)|facets]] of the [[steritruncated 6-orthoplex]].
| |
| | |
| === Images ===
| |
| {{5-simplex Coxeter plane graphs|t014|100}}
| |
| | |
| ==Stericantellated 5-simplex==
| |
| {| class="wikitable" align="right" style="margin-left:10px" width="280"
| |
| |-
| |
| |bgcolor=#e7dcc3 align=center colspan=3|'''Stericantellated 5-simplex'''
| |
| |-
| |
| |bgcolor=#e7dcc3|Type
| |
| |colspan=2|[[Uniform 5-polytope]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Schläfli symbol]]
| |
| |colspan=2|rr2r{3,3,3,3}
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
| |
| |colspan=2|{{CDD||node_1|3|node||3|node_1|3|node|3|node_1}}<BR>or {{CDD|node_1|split1|nodes|3ab|nodes_11}}
| |
| |-
| |
| |bgcolor=#e7dcc3|4-faces
| |
| | 62
| |
| |12 [[Cantellated 5-cell|rr{3,3,3}]]<BR>30 [[Cantellated tetrahedron|rr{3,3}]]x{}<BR>20 [[Duoprism|{3}×{3}]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Cells
| |
| |420
| |
| |60 [[Cantellated tetrahedron|rr{3,3}]]<BR>240 [[Triangular prism|{}×{3}]]<BR>90 [[Cube|{}×{}×{}]]<BR>30 [[Rectified tetrahedron|r{3,3}]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Faces
| |
| |900
| |
| |360 [[Triangle|{3}]]<BR>540 [[Square (geometry)|{4}]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Edges
| |
| |colspan=2|720
| |
| |-
| |
| |bgcolor=#e7dcc3|Vertices
| |
| |colspan=2|180
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Vertex figure]]
| |
| |colspan=2|[[File:Stericantellated 5-simplex verf.png|100px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter group]]
| |
| |colspan=2|A<sub>5</sub>×2, [<span/>[3,3,3,3]], order 1440
| |
| |-
| |
| |bgcolor=#e7dcc3|Properties
| |
| |colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
| |
| |}
| |
| | |
| === Alternate names ===
| |
| * Stericantellated hexateron
| |
| * Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)<ref>Klitizing, (x3o3x3o3x - card)</ref>
| |
| | |
| === Coordinates ===
| |
| The coordinates can be made in 6-space, as permutations of:
| |
| : (0,1,1,2,2,3)
| |
| | |
| This construction exists as one of 64 [[orthant]] [[Facet (geometry)|facets]] of the [[stericantellated 6-orthoplex]].
| |
| | |
| === Images ===
| |
| {{5-simplex2 Coxeter plane graphs|t024|100}}
| |
| | |
| ==Stericantitruncated 5-simplex==
| |
| | |
| {| class="wikitable" align="right" style="margin-left:10px" width="280"
| |
| |-
| |
| |bgcolor=#e7dcc3 align=center colspan=3|'''Stericantitruncated 5-simplex'''
| |
| |-
| |
| |bgcolor=#e7dcc3|Type
| |
| |colspan=2|[[Uniform 5-polytope]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Schläfli symbol]]
| |
| |colspan=2|t<sub>0,1,2,4</sub>{3,3,3,3}
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
| |
| |colspan=2|{{CDD|node_1|3|node_1||3|node_1|3|node|3|node_1}}
| |
| |-
| |
| |bgcolor=#e7dcc3|4-faces
| |
| |62
| |
| |
| |
| |-
| |
| |bgcolor=#e7dcc3|Cells
| |
| |480
| |
| |
| |
| |-
| |
| |bgcolor=#e7dcc3|Faces
| |
| |1140
| |
| |
| |
| |-
| |
| |bgcolor=#e7dcc3|Edges
| |
| |colspan=2|1080
| |
| |-
| |
| |bgcolor=#e7dcc3|Vertices
| |
| |colspan=2|360
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Vertex figure]]
| |
| |colspan=2|[[File:Stericanitruncated 5-simplex verf.png|100px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter group]]
| |
| |colspan=2|A<sub>5</sub> [3,3,3,3], order 720
| |
| |-
| |
| |bgcolor=#e7dcc3|Properties
| |
| |colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
| |
| |}
| |
| | |
| === Alternate names ===
| |
| * Stericantitruncated hexateron
| |
| * Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)<ref>Klitizing, (x3x3x3o3x - cograx)</ref>
| |
| | |
| === Coordinates ===
| |
| The coordinates can be made in 6-space, as 360 permutations of:
| |
| : (0,1,1,2,3,4)
| |
| | |
| This construction exists as one of 64 [[orthant]] [[Facet (geometry)|facets]] of the [[stericantitruncated 6-orthoplex]].
| |
| | |
| === Images ===
| |
| {{5-simplex Coxeter plane graphs|t0124|100}}
| |
| | |
| ==Steriruncitruncated 5-simplex==
| |
| {| class="wikitable" align="right" style="margin-left:10px" width="280"
| |
| |-
| |
| |bgcolor=#e7dcc3 align=center colspan=3|'''Steriruncitruncated 5-simplex'''
| |
| |-
| |
| |bgcolor=#e7dcc3|Type
| |
| |colspan=2|[[Uniform 5-polytope]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Schläfli symbol]]
| |
| |colspan=2|2t2r{3,3,3,3}
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
| |
| |colspan=2|{{CDD||node_1|3|node_1||3|node|3|node_1|3|node_1}}<BR>or {{CDD|node|split1|nodes_11|3ab|nodes_11}}
| |
| |-
| |
| |bgcolor=#e7dcc3|4-faces
| |
| |62
| |
| |12 [[Runcitruncated 5-cell|t<sub>0,1,3</sub>{3,3,3}]]<BR>30 {}×[[Truncated tetrahedron|t{3,3}]]<BR>20 [[Duoprism|{6}×{6}]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Cells
| |
| |450
| |
| |
| |
| |-
| |
| |bgcolor=#e7dcc3|Faces
| |
| |1110
| |
| |
| |
| |-
| |
| |bgcolor=#e7dcc3|Edges
| |
| |colspan=2|1080
| |
| |-
| |
| |bgcolor=#e7dcc3|Vertices
| |
| |colspan=2|360
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Vertex figure]]
| |
| |colspan=2|[[File:Steriruncitruncated 5-simplex verf.png|100px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter group]]
| |
| |colspan=2|A<sub>5</sub>×2, [<span/>[3,3,3,3]], order 1440
| |
| |-
| |
| |bgcolor=#e7dcc3|Properties
| |
| |colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
| |
| |}
| |
| | |
| === Alternate names ===
| |
| * Steriruncitruncated hexateron
| |
| * Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)<ref>Klitizing, (x3x3o3x3x - captid)</ref>
| |
| | |
| === Coordinates ===
| |
| The coordinates can be made in 6-space, as 360 permutations of:
| |
| : (0,1,2,2,3,4)
| |
| | |
| This construction exists as one of 64 [[orthant]] [[Facet (geometry)|facets]] of the [[steriruncitruncated 6-orthoplex]].
| |
| | |
| === Images ===
| |
| {{5-simplex2 Coxeter plane graphs|t0134|100}}
| |
| | |
| == Omnitruncated 5-simplex ==
| |
| {| class="wikitable" align="right" style="margin-left:10px" width="280"
| |
| |-
| |
| |bgcolor=#e7dcc3 align=center colspan=3|'''Omnitruncated 5-simplex'''
| |
| |-
| |
| |bgcolor=#e7dcc3|Type
| |
| |colspan=2|[[Uniform 5-polytope]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Schläfli symbol]]
| |
| |colspan=2|tr2r{3,3,3,3}
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
| |
| |colspan=2|{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>or {{CDD|node_1|split1|nodes_11|3ab|nodes_11}}
| |
| |-
| |
| |bgcolor=#e7dcc3|4-faces
| |
| |62
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| |12 [[Omnitruncated 5-cell|t<sub>0,1,2,3</sub>{3,3,3}]][[Image:Schlegel half-solid omnitruncated 5-cell.png|25px]]<BR>30 [[Truncated octahedral prism|{}×tr{3,3}]][[Image:Truncated octahedral prism.png|25px]]<BR>20 [[Duoprism|{6}×{6}]][[Image:6-6 duoprism.png|25px]]
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| |-
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| |bgcolor=#e7dcc3|Cells
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| |540
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| |360 [[truncated octahedron|t{3,4}]][[Image:Truncated octahedron.png|25px]]<BR>90 [[cube|{4,3}]][[Image:Tetragonal prism.png|25px]]<BR>90 [[hexagonal prism|{}×{6}]][[Image:Hexagonal prism.png|25px]]
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| |-
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| |bgcolor=#e7dcc3|Faces
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| |1560
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| |480 [[Hexagon|{6}]]<BR>1080 [[Square (geometry)|{4}]]
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| |-
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| |bgcolor=#e7dcc3|Edges
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| |colspan=2|1800
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| |-
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| |bgcolor=#e7dcc3|Vertices
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| |colspan=2|720
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| |-
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| |bgcolor=#e7dcc3|[[Vertex figure]]
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| |colspan=2|[[File:Omnitruncated 5-simplex verf.png|80px]]<BR>[[Irregular 5-cell]]
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group]]
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| |colspan=2| A<sub>5</sub>×2, [<span/>[3,3,3,3]], order 1440
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| |-
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| |bgcolor=#e7dcc3|Properties
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| |colspan=2|[[Convex polytope|convex]], [[isogonal]], [[zonotope]]
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| |}
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| The '''omnitruncated 5-simplex''' has 720 [[vertex (geometry)|vertices]], 1800 [[Edge (geometry)|edge]]s, 1560 [[Face (geometry)|faces]] (480 [[hexagon]]s and 1080 [[Square (geometry)|squares]]), 540 [[Cell (geometry)|cells]] (360 [[truncated octahedron]]s, 90 [[cube]]s, and 90 [[hexagonal prism]]s), and 62 [[4-face]]s (12 [[omnitruncated 5-cell]]s, 30 [[truncated octahedral prism]]s, and 20 6-6 [[duoprism]]s).
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| | |
| === Alternate names ===
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| * Steriruncicantitruncated 5-simplex (Full description of [[omnitruncation]] for 5-polytopes by Johnson)
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| * Omnitruncated hexateron
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| * Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers)<ref>Klitizing, (x3x3x3x3x - gocad)</ref>
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| | |
| === Coordinates ===
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| The vertices of the ''truncated 5-simplex'' can be most simply constructed on a [[hyperplane]] in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive [[orthant]] [[Facet (geometry)|facet]] of the [[steriruncicantitruncated 6-orthoplex]], t<sub>0,1,2,3,4</sub>{3<sup>4</sup>,4}, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}.
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| | |
| === Images ===
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| | |
| {| class=wikitable width=640
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| |- valign=top align=center
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| |
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| {{5-simplex2 Coxeter plane graphs|t01234|120}}
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| |[[Stereographic projection]]<BR>[[Image:Omnitruncated Hexateron.png|160px]]
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| |}
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| | |
| === Permutohedron ===
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| | |
| The omnitruncated 5-simplex is the permutohedron of order 6. It is also a [[zonotope]], the [[Minkowski sum]] of six line segments parallel to the six lines through the origin and the six vertices of the 5-simplex.
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| | |
| {| class=wikitable
| |
| |[[Image:Omnitruncated Hexateron as Permutohedron.svg|480px]]<BR>[[Orthogonal projection]], vertices labeled as a [[permutohedron]].
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| |}
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| | |
| === Related honeycomb ===
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| The [[omnitruncated 5-simplex honeycomb]] is constructed by '''omnitruncated 5-simplex''' facets with 3 [[Facet (geometry)|facets]] around each [[Ridge (geometry)|ridge]]. It has [[Coxeter-Dynkin diagram]] of {{CDD|branch_11|3ab|nodes_11|3ab|branch_11}}.
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| | |
| {| class=wikitable
| |
| |- align=center
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| ![[Coxeter group]]
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| !<math>{\tilde{I}}_{1}</math>
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| !<math>{\tilde{A}}_{2}</math>
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| !<math>{\tilde{A}}_{3}</math>
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| !<math>{\tilde{A}}_{4}</math>
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| !<math>{\tilde{A}}_{5}</math>
| |
| |- align=center
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| ![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
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| |{{CDD|node_1|infin|node_1}}
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| |{{CDD||branch_11|split2|node_1}}
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| |{{CDD|branch_11|3ab|branch_11}}
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| |{{CDD|branch_11|3ab|nodes_11|split2|node_1}}
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| |{{CDD|branch_11|3ab|nodes_11|3ab|branch_11}}
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| |-
| |
| !Picture
| |
| |[[File:Uniform apeirogon.png|100px]]
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| |[[File:Uniform tiling 333-t012.png|100px]]
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| |[[File:Bitruncated cubic honeycomb4.png|100px]]
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| |
| |
| |
| |
| |-
| |
| !Name
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| |[[Apeirogon]]
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| |[[Hextille]]
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| |[[Bitruncated cubic honeycomb|Omnitruncated<BR>3-simplex<BR>honeycomb]]
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| |[[Omnitruncated 4-simplex honeycomb|Omnitruncated<BR>4-simplex<BR>honeycomb]]
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| |[[Omnitruncated 5-simplex honeycomb|Omnitruncated<BR>5-simplex<BR>honeycomb]]
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| |-
| |
| !Facets
| |
| |[[File:Segment definition.svg|100px]]
| |
| |[[File:Omnitruncated 2-simplex graph.png|100px]]
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| |[[File:Omnitruncated 3-simplex.png|100px]]
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| |[[File:Omnitruncated 4-simplex.png|100px]]
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| |[[File:Omnitruncated 5-simplex.png|100px]]
| |
| |}
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| | |
| == Related uniform polytopes ==
| |
| | |
| These polytopes are a part of 19 [[Uniform_polyteron#The_A5_.5B3.2C3.2C3.2C3.5D_family_.285-simplex.29|uniform polytera]] based on the [3,3,3,3] [[Coxeter group]], all shown here in A<sub>5</sub> [[Coxeter plane]] [[orthographic projection]]s. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
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| | |
| {{Hexateron family}}
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| ==Notes==
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| {{reflist}}
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| | |
| == 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]
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| *** (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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| *** (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)}} x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad
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| == External links ==
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| * {{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
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| * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
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| * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
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| {{Polytopes}}
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| [[Category:5-polytopes]]
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