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| | They call me Emilia. I used to be unemployed but now I am a librarian and the wage has been truly satisfying. One of the things she enjoys most is to do aerobics and now she is trying to earn money with it. South Dakota is where me and my husband reside.<br><br>Feel free to visit my blog: [http://nuvem.tk/altergalactica/AliceedMaurermy nuvem.tk] |
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| |[[File:Schlegel_wireframe_5-cell.png|150px]]<BR>[[5-cell]]<BR>{{CDD|node|3|node|3|node|3|node_1}}
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| |[[Image:Schlegel half-solid truncated pentachoron.png|150px]]<BR>Truncated 5-cell<BR>{{CDD|node|3|node|3|node_1|3|node_1}}
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| |[[File:Schlegel_half-solid_bitruncated_5-cell.png|150px]]<BR>Bitruncated 5-cell<BR>{{CDD|node|3|node_1|3|node_1|3|node}}
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| |- align=center
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| |colspan=4|Schlegel diagrams centered on [3,3] (cells at opposite at [3,3])
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| |}
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| In [[geometry]], a '''truncated 5-cell''' is a [[uniform polychoron]] (4-dimensional uniform [[polytope]]) formed as the [[Truncation (geometry)|truncation]] of the regular [[5-cell]].
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| There are two degrees of trunctions, including a [[bitruncation]].
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| == Truncated 5-cell==
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| {| class="wikitable" align="right" style="margin-left:10px" width="250"
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| |-
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| !bgcolor=#e7dcc3 colspan=3|Truncated 5-cell
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| |-
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| |bgcolor=#ffffff align=center colspan=3|[[Image:Schlegel half-solid truncated pentachoron.png|250px]]<BR>[[Schlegel diagram]]<BR>([[tetrahedron]] cells visible)
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| |-
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| |bgcolor=#e7dcc3|Type
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| |colspan=2|[[Uniform polychoron]]
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]
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| |colspan=2|t<sub>0,1</sub>{3,3,3}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
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| |colspan=2|{{CDD|node_1|3|node_1|3|node|3|node}}
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| |-
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| |bgcolor=#e7dcc3|Cells
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| |10
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| |5 [[Tetrahedron|(''3.3.3'')]] [[Image:Tetrahedron.png|20px]]<BR>5 [[truncated tetrahedron|(''3.6.6'')]] [[Image:Truncated tetrahedron.png|20px]]
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| |-
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| |bgcolor=#e7dcc3|Faces
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| |30
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| |20 {3}<BR>10 {6}
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| |-
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| |bgcolor=#e7dcc3|Edges
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| |colspan=2|40
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| |-
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| |bgcolor=#e7dcc3|Vertices
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| |colspan=2|20
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| |-
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| |bgcolor=#e7dcc3|[[Vertex figure]]
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| |colspan=2|[[Image:Truncated 5-cell verf.png|80px]]<BR>Irr. tetrahedron
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
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| |colspan=2|A<sub>4</sub>, [3,3,3], order 120
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| |-
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| |bgcolor=#e7dcc3|Properties
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| |colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]]
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| |-
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| |bgcolor=#e7dcc3|Uniform index
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| |colspan=2|''[[Rectified 5-cell|2]]'' 3 ''[[Cantellated 5-cell|4]]''
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| |}
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| The '''truncated 5-cell''', '''truncated pentatope''' or '''truncated 4-simplex''' is bounded by 10 [[cell (geometry)|cells]]: 5 [[tetrahedron|tetrahedra]], and 5 [[truncated tetrahedron|truncated tetrahedra]]. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the [[vertex figure]] is an elongated tetrahedron.
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| ===Construction===
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| The truncated 5-cell may be constructed from the [[5-cell]] by [[Truncation (geometry)|truncating]] its vertices at 1/3 the edge length. This truncates the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned on the original vertices.
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| ===Structure===
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| The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces.
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| ===Projections===
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| The tetrahedron-first parallel projection of the truncated 5-cell into 3-dimensional space has the following structure:
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| * The projection envelope is a [[truncated tetrahedron]].
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| * One of the truncated tetrahedral cells project onto the entire envelope.
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| * One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope.
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| * Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells.
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| * Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining trucated tetrahedral cells.
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| This layout of cells in projection is analogous to the layout of faces in the face-first projection of the truncated tetrahedron into 2-dimensional space. The truncated 5-cell is the 4-dimensional analogue of the truncated tetrahedron.
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| === Images ===
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| {{4-simplex Coxeter plane graphs|t01|150}}
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| <gallery>
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| Image:Truncated pentachoron net.png|[[Net (polytope)|net]]
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| Image:Truncated simplex stereographic.png|[[stereographic projection]]<BR>(centered on [[truncated tetrahedron]])
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| </gallery>
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| === Alternate names ===
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| * Truncated pentatope
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| * Truncated [[Simplex|4-simplex]]
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| * Truncated pentachoron (Acronym: tip) (Jonathan Bowers)
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| ===Coordinates===
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| The [[Cartesian coordinate]]s for the vertices of an origin-centered truncated 5-cell having edge length 2 are:
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| {|
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| :<math>\left( \frac{3}{\sqrt{10}},\ \sqrt{3 \over 2},\ \pm\sqrt{3},\ \pm1\right)</math>
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| :<math>\left( \frac{3}{\sqrt{10}},\ \sqrt{3 \over 2},\ 0,\ \pm2\right)</math>
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| :<math>\left( \frac{3}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm2\right)</math>
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| :<math>\left( \frac{3}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{4}{\sqrt{3}},\ 0 \right)</math>
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| :<math>\left( \frac{3}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math>
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| :<math>\left( \frac{3}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math>
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| :<math>\left( -\sqrt{2 \over 5},\ \sqrt{2 \over 3},\ \frac{2}{\sqrt{3}},\ \pm2\right)</math>
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| :<math>\left( -\sqrt{2 \over 5},\ \sqrt{2 \over 3},\ \frac{-4}{\sqrt{3}},\ 0 \right)</math>
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| :<math>\left( -\sqrt{2 \over 5},\ -\sqrt{6},\ 0,\ 0 \right)</math>
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| :<math>\left( \frac{-7}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math>
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| :<math>\left( \frac{-7}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math>
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| :<math>\left( \frac{-7}{\sqrt{10}},\ -\sqrt{3 \over 2},\ 0,\ 0 \right)</math>
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| |}
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| More simply, the vertices of the ''truncated 5-cell'' can be constructed on a [[hyperplane]] in 5-space as permutations of (0,0,0,1,2) ''or'' of (0,1,2,2,2). These coordinates come from positive [[orthant]] facets of the [[truncated pentacross]] and [[bitruncated penteract]] respectively.
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| == Bitruncated 5-cell==
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| {| class="wikitable" align="right" style="margin-left:10px" width="250"
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| |-
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| |bgcolor=#ffffff align=center colspan=3|[[Image:Schlegel half-solid bitruncated 5-cell.png|220px]]<BR>[[Schlegel diagram]] with alternate cells hidden.
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| |-
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| |bgcolor=#e7dcc3 align=center colspan=3|Bitruncated 5-cell
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| |-
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| |bgcolor=#e7dcc3|Type
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| |colspan=2|[[Uniform polychoron]]
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]
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| |colspan=2| t<sub>1,2</sub>{3,3,3}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
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| |colspan=2|{{CDD|node|3|node_1|3|node_1|3|node}}<BR>or {{CDD|branch_11|3ab|nodes}}
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| |-
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| |bgcolor=#e7dcc3|Cells
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| |colspan=2|10 ([[Truncated tetrahedron|''3.6.6'']]) [[Image:Truncated tetrahedron.png|20px]]
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| |-
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| |bgcolor=#e7dcc3|Faces
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| |40
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| |20 {3}<br/>20 {6}
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| |-
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| |bgcolor=#e7dcc3|Edges
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| |colspan=2|60
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| |-
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| |bgcolor=#e7dcc3|Vertices
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| |colspan=2|30
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| |-
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| |bgcolor=#e7dcc3 valign=center|[[Vertex figure]]
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| |colspan=2|[[Image:Bitruncated 5-cell verf.png|80px]]<BR>([[Tetragonal disphenoid]])
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
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| |colspan=2|A<sub>4</sub>, <nowiki>[[3,3,3]]</nowiki>, order 240
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| |-
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| |bgcolor=#e7dcc3|Properties
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| |colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]] [[isotoxal figure|isotoxal]], [[Isochoric figure|isochoric]]
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| |-
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| |bgcolor=#e7dcc3|Uniform index
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| |colspan=2|''[[Runcinated pentachoron|5]]'' 6 ''[[Cantitruncated 5-cell|7]]''
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| |}
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| The '''[[bitruncation (geometry)|bitruncated]] [[5-cell]]''' (also called a '''bitruncated pentachoron''', '''decachoron''' and '''10-cell''') is a 4-dimensional [[polytope]], or [[polychoron]], composed of 10 [[cell (mathematics)|cells]] in the shape of [[truncated tetrahedron|truncated tetrahedra]]. Each hexagonal face of the truncated tetrahedra is joined in complementary orientation to the neighboring truncated tetrahedron. Each edge is shared by two hexagons and one triangle. Each vertex is surrounded by 4 truncated tetrahedral cells in a [[tetragonal disphenoid]] [[vertex figure]].
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| The bitruncated 5-cell is the [[intersection (set theory)|intersection]] of two [[Pentachoron|pentachora]] in dual configuration. As such, it is also the intersection of a [[penteract]] with the hyperplane that bisects the penteract's long diagonal orthogonally. In this sense it is the 4-dimensional analog of the [[Octahedron|regular octahedron]] (intersection of regular tetrahedra in dual configuration / [[tesseract]] bisection on long diagonal) and the regular hexagon (equilateral triangles / cube). The 5-dimensional analog is the [[birectified 5-simplex]], and the <math>n</math>-dimensional analog is the polytope whose [[Coxeter-Dynkin diagram|Coxeter–Dynkin diagram]] is linear with rings on the middle one or two nodes.
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| The bitruncated 5-cell is one of the two non-regular [[uniform polychoron|uniform polychora]] which are [[cell-transitive]]. The other is the [[bitruncated 24-cell]], which is composed of 48 truncated cubes.
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| === Symmetry ===
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| This polychoron has a higher extended pentachoric symmetry (A<sub>4</sub>, <nowiki>[[3,3,3]]</nowiki>), doubled to order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.
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| === Alternative names ===
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| * Bitruncated 5-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]])
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| * 10-cell as a [[Isochoric figure|cell-transitive]] 4-polytope
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| * Bitruncated pentachoron
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| * Bitruncated pentatope
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| * Bitruncated [[Simplex|4-simplex]]
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| * Decachoron (Acronym: deca) (Jonathan Bowers)
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| === Images ===
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| {{4-simplex2 Coxeter plane graphs|t12|150}}
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| {| class=wikitable width=440
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| |[[File:Decachoron stereographic (hexagon).png|220px]]<BR>stereographic projection of spherical polychoron <br>(centred on a hexagon face)
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| |[[File:Bitruncated 5-cell net.png|220px]]<BR>[[Net (polytope)]]
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| |}
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| === Coordinates ===
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| The [[Cartesian coordinate]]s of an origin-centered bitruncated 5-cell having edge length 2 are:
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| :<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{5}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ 0\right)</math>
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| :<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{5}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm1\right)</math>
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| :<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{4}{\sqrt{3}},\ 0\right)</math>
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| :<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ \pm2\right)</math>
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| :<math>\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm1\right)</math>
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| :<math>\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\ \pm2\right)</math>
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| :<math>\pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{4}{\sqrt{3}},\ 0\right)</math>
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| :<math>\pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ \pm2\right)</math>
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| More simply, the vertices of the bitruncated 5-cell can be constructed on a [[hyperplane]] in 5-space as permutations of (0,0,1,2,2). These represent positive [[orthant]] facets of the [[bitruncated pentacross]].
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| Another 5-space construction are all 20 permutations of (1,0,0,0,-1).
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| ==== Related polytopes====
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| {{Isotopic uniform simplex polytopes}}
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| == Related regular skew polyhedron ==
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| The [[regular skew polyhedron]], {6,4|3}, exists in 4-space with 4 hexagonal around each vertex, in a zig-zagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 20 vertices. The 20 triangular faces of the bitruncated 5-cell can be seen as removed. The dual regular skew polyhedron, {4,6|3}, is similarly related to the square faces of the [[runcinated 5-cell]].
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| === Related polytopes ===
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| These polytope are from a set of 9 [[Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[Coxeter group]].
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| {{Pentachoron family}}
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| == References ==
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| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
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| ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
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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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| *[[Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, ISBN 0-486-40919-8 p.88 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)
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| **Coxeter, H. S. M. ''Regular Skew Polyhedra in Three and Four Dimensions.'' Proc. London Math. Soc. 43, 33-62, 1937.
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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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| *{{GlossaryForHyperspace | anchor=Pentachoron | title=Pentachoron}}
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| ** {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron - Model 3}}
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| * {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} x3x3o3o - tip, o3x3x3o - deca
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| {{Polytopes}}
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora]]
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