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| {{For|the sequence of fifth element numbers of Pascal's triangle|Pentatope number}}
| | Skull accessories are certainly very popular among celebrities like Angelina Jolie, Lindsay Lohan, Johnny Depp for example. They have been photographed a number times wearing skull decorations. This has also helped to make skull accessories a fashion phenomenon.<br><br> |
| {{Infobox polychoron |
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| Name=Regular 5-cell<BR>(pentachoron)<BR>(4-simplex)|
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| Image_File=Schlegel wireframe 5-cell.png|
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| Image_Caption=[[Schlegel diagram]]<BR>(vertices and edges)|
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| Type=[[Convex regular 4-polytope]]|
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| Family=[[Simplex]]|
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| Last= |
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| Index=1|
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| Next=[[Rectified_5-cell|2]]|
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| Schläfli={3,3,3}|
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| CD={{CDD|node_1|3|node|3|node|3|node}}|
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| Cell_List=5 [[tetrahedron|{3,3}]] [[Image:3-simplex_t0.svg|20px]] |
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| Face_List= 10 {3} [[Image:2-simplex_t0.svg|20px]]|
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| Edge_Count= 10|
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| Vertex_Count= 5|
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| Petrie_Polygon=[[pentagon]]|
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| Coxeter_Group= A<sub>4</sub>, [3,3,3]|
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| Vertex_Figure=[[Image:5-cell verf.png|80px]]<BR>([[tetrahedron]])|
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| Dual=[[Self-dual polytope|Self-dual]]|
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| Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]]
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| }}
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| [[Image:tetrahedron.jpg|thumb|right|Vertex figure: ''tetrahedron'']]
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| In [[geometry]], the '''5-cell''' is a [[four-dimensional space|four-dimensional]] object bounded by 5 [[tetrahedron|tetrahedral cells]]. It is also known as the '''pentachoron''', '''pentatope''', or [[#Irregular 5-cell|'''tetrahedral hyperpyramid''']]. It is a '''4-[[simplex]]''', the simplest possible [[convex regular 4-polytope]] (four-dimensional analogue of a [[polyhedron]]), and is analogous to the [[tetrahedron]] in three dimensions and the [[triangle]] in two dimensions.
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| The '''regular 5-cell''' is bounded by [[regular tetrahedron|regular tetrahedra]], and is one of the six [[regular convex polychoron|regular convex polychora]], represented by [[Schläfli symbol]] {3,3,3}.
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| ==Geometry==
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| The 5-cell is [[Self-dual polytope|self-dual]], and its [[vertex figure]] is a tetrahedron. Its maximal intersection with 3-dimensional space is the [[triangular prism]]. Its [[dihedral angle]] is cos<sup>−1</sup>(1/4), or approximately 75.52°.
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| ===Construction===
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| The 5-cell can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. (The 5-cell is essentially a 4-dimensional [[Pyramid (geometry)|pyramid]] with a [[Tetrahedron|tetrahedral]] base.)
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| The [[Cartesian coordinates]] of the vertices of an origin-centered regular 5-cell having edge length 2 are:
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| :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math>
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| :<math>\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math>
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| :<math>\left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)</math>
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| :<math>\left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right)</math>
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| Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2√2:
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| :(1,1,1,-1), (1,-1,-1,-1), (-1,1,-1,-1), (-1,-1,1,-1), (0,0,0,√5 - 1)
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| The vertices of a 4-simplex (with edge √2) can be more simply constructed on a [[hyperplane]] in 5-space, as permutations of (0,0,0,0,1) ''or'' (0,1,1,1,1); in these positions it is a [[facet (geometry)|facet]] of, respectively, the [[5-orthoplex]] or the [[rectified penteract]].
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| ===Projections===
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| The A<sub>4</sub> Coxeter plane projects the 5-cell into a regular [[pentagon]] and [[pentagram]].
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| {{4-simplex Coxeter plane graphs|t0|150}}
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| {|class="wikitable" width=640
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| !colspan=2|Projections to 3 dimensions
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| |- valign=top align=center
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| |[[Image:Stereographic polytope 5cell.png|240px]]<BR>[[Stereographic projection]] wireframe (edge projected onto a [[3-sphere]])
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| |[[Image:5-cell.gif]]<BR>A 3D projection of a 5-cell performing a [[SO(4)#Geometry of 4D rotations|simple rotation]]
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| |- valign=top align=center
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| |[[Image:Pentatope-vertex-first-small.png]]<BR>The vertex-first projection of the pentachoron into 3 dimensions has a [[tetrahedron|tetrahedral]] projection envelope. The closest vertex of the pentachoron projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex.
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| |[[Image:5cell-edge-first-small.png]]<BR>The edge-first projection of the pentachoron into 3 dimensions has a [[triangular dipyramid|triangular dipyramidal]] envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope.
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| |- valign=top align=center
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| |[[Image:5cell-face-first-small.png]]<BR>The face-first projection of the pentachoron into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face projects to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection.
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| |[[Image:5cell-cell-first-small.png|320px]]<BR>The cell-first projection of the pentachoron into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here.
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| |}
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| == Alternative names ==
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| * Pentachoron
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| * [[Simplex|4-simplex]]
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| * Pentatope
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| * Pentahedroid (Henry Parker Manning)
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| * Pen ([[Jonathan Bowers]]: for pentachoron)
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| * Hyperpyramid
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| == Related polytopes and honeycomb ==
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| The pentachoron (5-cell) is the simplest of 9 [[Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[Coxeter group]].
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| {{Pentachoron family}}
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| {{1 k2 polytopes}}
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| {{2 k1 polytopes}}
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| It is in the sequence of [[regular polychora]]: the [[tesseract]] {4,3,3}, [[120-cell]] {5,3,3}, of Euclidean 4-space, and [[hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space. All of these have a [[tetrahedron|tetrahedral]] [[vertex figure]].
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| {{Tetrahedral vertex figure tessellations}}
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| It is similar to three [[regular polychora]]: the [[tesseract]] {4,3,3}, [[600-cell]] {3,3,5} of Euclidean 4-space, and the [[order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. All of these have a [[tetrahedron|tetrahedral]] cells.
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| {{Tetrahedral cell tessellations}}
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| == Irregular 5-cell ==
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| [[File:5-celled verfs.png|thumb|100|Uniform 5-polytope vertex figures]]
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| The '''tetrahedral pyramid''' is a special case of a '''5-cell''', a [[polyhedral pyramid]], constructed as a regular [[tetrahedron]] base in a 3-space [[hyperplane]], and an [[Apex (geometry)|apex]] point ''above'' the hyperplane. The four ''sides'' of the pyramid are made of tetrahedron cells.
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| Some [[uniform 5-polytope]]s have '''tetrahedral pyramid''' [[vertex figure]]s:
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| {| class=wikitable
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| |+ Symmetry {{CDD|node|3|node|3|node}}, [3,3], (*332)
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| |-
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| ![[Schlegel diagram|Schlegel<BR>diagram]]
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| |[[File:5-cell prism verf.png|100px]]
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| |[[File:Tesseractic prism verf.png|100px]]
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| |[[File:120-cell prism verf.png|100px]]
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| |[[File:Truncated 5-simplex verf.png|100px]]
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| |[[File:Truncated 5-cube verf.png|100px]]
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| |[[File:Truncated_24-cell_honeycomb_verf.png|100px]]
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| |-
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| !Name<BR>[[Coxeter diagram|Coxeter<BR>diagram]]
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| ![[5-cell prism|{ }×{3,3,3}]]<BR>{{CDD|node_1|2|node_1|3|node|3|node|3|node}}
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| ![[Tesseractic prism|{ }×{4,3,3}]]<BR>{{CDD|node_1|2|node_1|4|node|3|node|3|node}}
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| ![[120-cell prism|{ }×{5,3,3}]]<BR>{{CDD|node_1|2|node_1|5|node|3|node|3|node}}
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| ![[Truncated 5-simplex|t{3,3,3,3}]]<BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node}}
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| ![[Truncated 5-cube|t{4,3,3,3}]]<BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node}}
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| ![[Truncated 24-cell honeycomb|t{3,4,3,3}]]<BR>{{CDD|node_1|3|node_1|4|node|3|node|3|node}}
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| |}
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| Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a [[uniform polytope]] is represented by removing the ringed nodes of the Coxeter diagram.
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| {| class=wikitable
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| !Symmetry
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| !colspan=2|{{CDD|node|2|node|3|node}}, [2,3], (*223)
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| !colspan=2|{{CDD|node|3|node}}, [3], (*33)
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| ![2<sup>+</sup>,4], (2*2)
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| !{{CDD|node|2|node}}, [2], (*22)
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| |-
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| ![[Schlegel diagram|Schlegel<BR>diagram]]
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| |[[File:Bitruncated 5-simplex verf.png|100px]]
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| |[[File:Bitruncated penteract verf.png|100px]]
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| |[[File:Canitruncated 5-simplex verf.png|100px]]
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| |[[File:Canitruncated_5-cube_verf.png|100px]]
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| |[[File:Bicanitruncated 5-simplex verf.png|100px]]
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| |[[File:Bicanitruncated_5-cube_verf.png|100px]]
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| |-
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| !Name<BR>[[Coxeter diagram|Coxeter<BR>diagram]]
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| ![[bitruncated 5-simplex|t<sub>12</sub>α<sub>5</sub>]]<BR>{{CDD|node|3|node_1|3|node_1|3|node|3|node}}
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| ![[bitruncated 5-cube|t<sub>12</sub>γ<sub>5</sub>]]<BR>{{CDD|node|4|node_1|3|node_1|3|node|3|node}}
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| ![[Cantitruncated 5-simplex|t<sub>012</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node|3|node}}
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| ![[Cantitruncated 5-cube|t<sub>012</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node|3|node}}
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| ![[Bicantitruncated 5-simplex|t<sub>123</sub>α<sub>5</sub>]]<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node}}
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| ![[Bicantitruncated 5-cube|t<sub>123</sub>γ<sub>5</sub>]]<BR>{{CDD|node|4|node_1|3|node_1|3|node_1|3|node}}
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| |}
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| {| class=wikitable
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| !Symmetry
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| !colspan=3|{{CDD|node}}, [ ], (*)
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| ![2]<sup>+</sup>, (22)
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| ![ ]<sup>+</sup>, (1)
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| |-
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| ![[Schlegel diagram|Schlegel<BR>diagram]]
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| |[[File:Runcicantitruncated 5-simplex verf.png|100px]]
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| |[[File:Runcicantitruncated 5-cube verf.png|100px]]
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| |[[File:Runcicantitruncated 5-orthoplex verf.png|100px]]
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| |[[File:Omnitruncated 5-simplex verf.png|100px]]
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| |[[File:Omnitruncated 5-cube verf.png|100px]]
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| |-
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| !Name<BR>[[Coxeter diagram|Coxeter<BR>diagram]]
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| ![[Runcicantitruncated 5-simplex|t<sub>0123</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node}}
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| ![[Runcicantitruncated 5-cube|t<sub>0123</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node}}
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| ![[Runcicantitruncated 5-orthoplex|t<sub>0123</sub>β<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|4|node}}
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| ![[Omnitruncated 5-simplex|t<sub>01234</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
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| ![[Omnitruncated 5-cube|t<sub>01234</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1}}
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| |}
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| == References ==
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| * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
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| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
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| ** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
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| ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
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| ** '''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]
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| *** (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]
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| * [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1<sub>n1</sub>)
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| * [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
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| ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
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| ==External links==
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| * {{MathWorld|urlname=Pentatope|title=Pentatope}}
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| *{{GlossaryForHyperspace | anchor=Pentachoron | title=Pentachoron}}
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| ** {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron - Model 1}}
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| * {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x3o3o3o - pen}}
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| * [http://www.polytope.de/c5.html Der 5-Zeller (5-cell)] Marco Möller's Regular polytopes in R<sup>4</sup> (German)
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| * [http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, Regular polychora]
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| * [http://public.beuth-hochschule.de/~meiko/pentatope.html Java3D Applets]
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| {{4D regular polytopes}}
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| {{Polytopes}}
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora| 005]]
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