Multivalued dependency: Difference between revisions

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{| class=wikitable align=right width=450
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|- align=center valign=top
|[[File:4-simplex t0.svg|150px]]<BR>[[5-cell]]<BR>{{CDD|node_1|3|node|3|node|3|node}}
|[[File:4-simplex t02.svg|150px]]<BR>Cantellated 5-cell<BR>{{CDD|node_1|3|node|3|node_1|3|node}}
|[[File:4-simplex t012.svg|150px]]<BR>Cantitruncated 5-cell<BR>{{CDD|node_1|3|node_1|3|node_1|3|node}}
|-
!colspan=3|[[Orthogonal projection]]s in A<sub>4</sub> [[Coxeter plane]]
|}
In four-dimensional [[geometry]], a '''cantellated 5-cell''' is a convex [[uniform polychoron]], being a [[cantellation]] (a 2nd order truncation) of the regular [[5-cell]].
 
There are 2 unique degrees of runcinations of the 5-cell including with permutations truncations.
 
{{TOC left}}
{{-}}
==Cantellation 5-cell==
 
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 align=center colspan=3|Cantellated 5-cell
|-
|align=center colspan=3|[[Image:Schlegel half-solid cantellated 5-cell.png|220px]]<BR>[[Schlegel diagram]] with<BR>octahedral cells shown
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| t<sub>0,2</sub>{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD|node_1|3|node|3|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|20
|5 [[Image:cuboctahedron.png|20px]][[cuboctahedron|(3.4.3.4)]]<BR>5 [[Image:octahedron.png|20px]][[octahedron|(3.3.3.3)]]<BR>10 [[Image:triangular prism.png|20px]][[triangular prism|(3.4.4)]]
|-
|bgcolor=#e7dcc3|Faces
|80
|50[[Triangle|{3}]]<BR>30[[Square (geometry)|{4}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|90
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|30
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Cantellated 5-cell verf.png|80px]]<BR>Irreg. [[triangular prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|A<sub>4</sub>, [3,3,3], order 120
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Truncated 5-cell|3]]'' 4 ''[[Runcinated pentachoron|5]]''
|}
The '''[[Cantellation (geometry)|cantellated]] [[5-cell]]''' is a [[uniform polychoron]]. It has 30 vertices, 90 edges, 80 faces, and 20 cells. The cells are 5 [[cuboctahedron|cuboctahedra]], 5 [[octahedron|octahedra]], and 10 [[triangular prism]]s. Each vertex is surrounded by 2 cuboctahedra, 2 triangular prisms, and 1 octahedron; the [[vertex figure]] is a nonuniform triangular prism.
 
=== Alternate names ===
* Cantellated pentachoron
* Cantellated [[Simplex|4-simplex]]
* (small) prismatodispentachoron
* Rectified dispentachoron
* Small rhombated pentachoron (Acronym: Srip) (Jonathan Bowers)
 
=== Images ===
{{4-simplex Coxeter plane graphs|t02|150}}
 
{| class=wikitable width=480
|-
|[[File:Cantel_pentachoron1.png|160px]]<BR>Wireframe
|[[File:Cantel_pentachoron2.png|160px]]<BR>Ten [[triangular prism]]s colored green
|[[File:Cantel_pentachoron3.png|160px]]<BR>Five [[octahedra]] colored blue
|}
 
=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of the origin-centered cantellated 5-cell having edge length&nbsp;2 are:
 
{| class=wikitable
|
:<math>\left(2\sqrt{\frac{2}{5}},\  2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(2\sqrt{\frac{2}{5}},\  2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)</math>
:<math>\left(2\sqrt{\frac{2}{5}},\  0,\                  \pm\sqrt{3},\        \pm1\right)</math>
:<math>\left(2\sqrt{\frac{2}{5}},\  0,\                  0,\                  \pm2\right)</math>
:<math>\left(2\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\  0\right)</math>
:<math>\left(2\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm1\right)</math>
:<math>\left(\frac{-1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\  \pm\sqrt{3},\        \pm1\right)</math>
|
:<math>\left(\frac{-1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\  0,\                  \pm2\right)</math>
:<math>\left(\frac{-1}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left(\frac{-1}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right)</math>
:<math>\left(\frac{-1}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(\frac{-1}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)</math>
:<math>\left(-3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\  \frac{2}{\sqrt{3}},\  0\right)</math>
:<math>\left(-3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm1\right)</math>
|}
 
The vertices of the ''cantellated 5-cell'' can be most simply positioned in 5-space as permutations of:
: (0,0,1,1,2)
 
This construction is from the positive [[orthant]] facet of the [[cantellated 5-orthoplex]].
 
== Cantitruncated 5-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 align=center colspan=3|Cantitruncated 5-cell
|-
|align=center colspan=3|[[Image:Schlegel half-solid cantitruncated 5-cell.png|240px]]<BR>[[Schlegel diagram]] with Truncated tetrahedral cells shown
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| t<sub>0,1,2</sub>{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD|node_1|3|node_1|3|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|20
|5 [[Image:truncated octahedron.png|20px]][[truncated octahedron|(4.6.6)]]<BR>10 [[Image:triangular prism.png|20px]][[triangular prism|(3.4.4)]]<BR>&nbsp;5 [[Image:truncated tetrahedron.png|20px]][[truncated tetrahedron|(3.6.6)]]
|-
|bgcolor=#e7dcc3|Faces
|80
|20{3}<BR>30{4}<BR>30{6}
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|120
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|60
|-
|bgcolor=#e7dcc3 valign=center|[[Vertex figure]]
|colspan=2|[[Image:Cantitruncated 5-cell verf.png|80px]]<BR>[[Sphenoid (geometry)|sphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|A<sub>4</sub>, [3,3,3], order 120
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Bitruncated 5-cell|6]]'' 7 ''[[Runcitruncated 5-cell|8]]''
|}
The '''cantitruncated [[5-cell]]''' is a [[uniform polychoron]]. It is composed of 60 vertices, 120 edges, 80 faces, and 20 cells. The cells are: 5 [[truncated octahedron|truncated octahedra]], 10 [[triangular prism]]s, and 5 [[truncated tetrahedron|truncated tetrahedra]]. Each vertex is surrounded by 2 truncated octahedra, one triangular prism, and one truncated tetrahedron.
 
=== Alternative names ===
* Cantitruncated pentachoron
* Cantitruncated [[Simplex|4-simplex]]
* Great prismatodispentachoron
* Truncated dispentachoron
* Great rhombated pentachoron (Acronym: grip) (Jonathan Bowers)
 
=== Images ===
{{4-simplex Coxeter plane graphs|t012|150}}
 
{| class=wikitable width=480
|[[File:Cantitruncated_5_cell.png|240px]]<BR>[[Stereographic projection]] with its 10 [[triangular prism]]s.
|}
 
=== Cartesian coordinates ===
 
The [[Cartesian coordinate]]s of an origin-centered cantitruncated 5-cell having edge length&nbsp;2 are:
 
{| class=wikitable
|
:<math>\left(3\sqrt{\frac{2}{5}},\  \pm\sqrt{6},\          \pm\sqrt{3},\        \pm1\right)</math>
:<math>\left(3\sqrt{\frac{2}{5}},\  \pm\sqrt{6},\          0,\                  \pm2\right)</math>
:<math>\left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm3\right)</math>
:<math>\left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm2\right)</math>
:<math>\left(\frac{1}{\sqrt{10}},\  \frac{5}{\sqrt{6}},\  \frac{5}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(\frac{1}{\sqrt{10}},\  \frac{5}{\sqrt{6}},\  \frac{-1}{\sqrt{3}},\ \pm3\right)</math>
:<math>\left(\frac{1}{\sqrt{10}},\  \frac{5}{\sqrt{6}},\  \frac{-4}{\sqrt{3}},\ \pm2\right)</math>
:<math>\left(\frac{1}{\sqrt{10}},\  -\sqrt{\frac{3}{2}},\  \sqrt{3},\            \pm3\right)</math>
:<math>\left(\frac{1}{\sqrt{10}},\  -\sqrt{\frac{3}{2}},\  -2\sqrt{3},\          0\right)</math>
:<math>\left(\frac{1}{\sqrt{10}},\  \frac{-7}{\sqrt{6}},\  \frac{2}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left(\frac{1}{\sqrt{10}},\  \frac{-7}{\sqrt{6}},\  \frac{-4}{\sqrt{3}},\ 0\right)</math>
:<math>\left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\  \frac{4}{\sqrt{3}},\  \pm2\right)</math>
|
:<math>\left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\  \frac{1}{\sqrt{3}},\  \pm3\right)</math>
:<math>\left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\  \frac{-5}{\sqrt{3}},\ \pm1\right)</math>
:<math>\left(-2\sqrt{\frac{2}{5}},\ 0,\                    \sqrt{3},\            \pm3\right)</math>
:<math>\left(-2\sqrt{\frac{2}{5}},\ 0,\                    -2\sqrt{3},\          0\right)</math>
:<math>\left(-2\sqrt{\frac{2}{5}},\ -4\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(-2\sqrt{\frac{2}{5}},\ -4\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)</math>
:<math>\left(\frac{-9}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\  \pm\sqrt{3},\        \pm1\right)</math>
:<math>\left(\frac{-9}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\  0,\                  \pm2\right)</math>
:<math>\left(\frac{-9}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\  \frac{2}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left(\frac{-9}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\  \frac{-4}{\sqrt{3}},\ 0\right)</math>
:<math>\left(\frac{-9}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\  \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(\frac{-9}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ 0\right)</math>
|}
 
These vertices can be more simply constructed on a [[hyperplane]] in 5-space, as the [[permutation]]s of:
: (0,0,1,2,3)
 
This construction is from the positive [[orthant]] [[facet (geometry)|facet]] of the [[cantitruncated 5-orthoplex]].
 
 
== Related polychora ==
These polytopes are art of a set of 9 [[Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[Coxeter group]].
{{Pentachoron family}}
 
== 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. (1966)
* {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron - Model 4, 7}}
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} x3o3x3o - srip, x3x3x3o - grip
 
{{Polytopes}}
 
[[Category:Four-dimensional geometry]]
[[Category:Polychora]]

Latest revision as of 13:54, 18 July 2014

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