View source for Rectified 5-cell

{{Short description|Uniform polychoron}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Rectified 5-cell'''
|-
|bgcolor=#ffffff align=center colspan=3|[[Image:Schlegel half-solid rectified 5-cell.png|220px]]<BR>[[Schlegel diagram]] with the 5 tetrahedral cells shown.
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| t<sub>1</sub>{3,3,3} or r{3,3,3}<BR>{3<sup>2,1</sup>} = <math>\left\{\begin{array}{l}3\\3,3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD|node|3|node_1|3|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|10
|5 [[tetrahedron|{3,3}]] [[Image:Tetrahedron.png|20px]]<BR>5 [[octahedron|''3.3.3.3'']] [[Image:Uniform polyhedron-33-t1.svg|20px]]
|-
|bgcolor=#e7dcc3|Faces
|colspan=2|30 [[Triangle|{3}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|30
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|10
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Rectified 5-cell verf.png|80px]]<BR>[[Triangular prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|A<sub>4</sub>, [3,3,3], order 120
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]
|colspan=2|[[Pentagon]]
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Pentachoron|1]]'' 2 ''[[Truncated 5-cell|3]]''
|}
In [[Four-dimensional space|four-dimensional]] [[geometry]], the '''[[Rectification (geometry)|rectified]] [[5-cell]]''' is a [[uniform 4-polytope]] composed of 5 regular tetrahedral and 5 regular octahedral [[cell (mathematics)|cells]]. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the [[vertex figure]] is a [[triangular prism]].

Topologically, under its highest symmetry, [3,3,3], there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra (which is geometrically the same as a regular octahedron). It is also topologically identical to a tetrahedron-octahedron segmentochoron.{{clarify|reason=What is a segmentochoron?|date=March 2020}}

The [[vertex figure]] of the ''rectified 5-cell'' is a uniform [[triangular prism]], formed by three [[octahedron|octahedra]] around the sides, and two [[tetrahedron|tetrahedra]] on the opposite ends.<ref>Conway, 2008</ref>

Despite having the same number of vertices as cells (10) and the same number of edges as faces (30), the rectified 5-cell is not self-dual because the vertex figure (a uniform triangular prism) is not a dual of the polychoron's cells.

== Wythoff construction ==
Seen in a [[configuration (polytope)|configuration matrix]], all incidence counts between elements are shown. The diagonal [[f-vector]] numbers are derived through the [[Wythoff construction]], dividing the full group order of a subgroup order by removing one mirror at a time.<ref>{{KlitzingPolytopes|../incmats/cont.htm|o3x4o3o - rap}}</ref>
{| class=wikitable
!A<sub>4</sub>||{{CDD|node|3|node_1|3|node|3|node}}
! [[k-face|''k''-face]]||f<sub>k</sub>||f<sub>0</sub>||colspan=1|f<sub>1</sub>||colspan=2|f<sub>2</sub>||colspan=2|f<sub>3</sub>||[[vertex figure|''k''-figure]]
!Notes
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node|2|node_x|2|node|3|node}}|| ( )
!f<sub>0</sub>
|BGCOLOR="#ffe0e0"|'''10'''||6||3||6||3||2||[[triangular prism|{3}x{ }]] || A<sub>4</sub>/A<sub>2</sub>A<sub>1</sub> = 5!/3!/2 = 10
|- align=right
|A<sub>1</sub>A<sub>1</sub> ||{{CDD|node_x|2|node_1|2|node_x|2|node}}|| { }
!f<sub>1</sub>
||2||BGCOLOR="#ffffe0"|'''30'''||1||2||2||1||[[Isosceles triangle|{ }v( )]] || A<sub>4</sub>/A<sub>1</sub>A<sub>1</sub> = 5!/2/2 = 30
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node|3|node_1|2|node_x|2|node}}|| rowspan=2|[[triangle|{3}]]
!rowspan=2|f<sub>2</sub>
||3||3||BGCOLOR="#e0ffe0"|'''10'''||BGCOLOR="#e0ffe0"|*||2||0||rowspan=2|{ } || A<sub>4</sub>/A<sub>2</sub>A<sub>1</sub> = 5!/3!/2 = 10
|- align=right
|A<sub>2</sub> ||{{CDD|node_x|2|node_1|3|node|2|node_x}}
||3||3||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''20'''||1||1 || A<sub>4</sub>/A<sub>2</sub> = 5!/3! = 20
|- align=right
|A<sub>3</sub> ||{{CDD|node|3|node_1|3|node|2|node_x}}|| [[octahedron|r{3,3}]]
!rowspan=2|f<sub>3</sub>
 ||6||12||4||4||BGCOLOR="#e0ffff"|'''5'''||BGCOLOR="#e0ffff"|*||rowspan=2|( ) || rowspan=2|A<sub>4</sub>/A<sub>3</sub> = 5!/4! = 5
|- align=right
|A<sub>3</sub> ||{{CDD|node_x|2|node_1|3|node|3|node}}|| [[tetrahedron|{3,3}]]
||4||6||0||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''5''' 
|}

== Structure==
Together with the simplex and [[24-cell]], this shape and its [[Dual polyhedron|dual]] (a polytope with ten vertices and ten [[triangular bipyramid]] facets) was one of the first 2-simple 2-simplicial 4-polytopes known. This means that all of its two-dimensional faces, and all of the two-dimensional faces of its dual, are triangles. In 1997, Tom Braden found another dual pair of examples, by gluing two rectified 5-cells together; since then, infinitely many 2-simple 2-simplicial polytopes have been constructed.<ref>{{citation
 | last1 = Eppstein | first1 = David | author1-link = David Eppstein
 | last2 = Kuperberg | first2 = Greg | author2-link = Greg Kuperberg
 | last3 = Ziegler | first3 = Günter M. | author3-link = Günter M. Ziegler
 | editor-last = Bezdek | editor-first = Andras
 | arxiv = math.CO/0204007
 | contribution = Fat 4-polytopes and fatter 3-spheres
 | volume = 253
 | pages = 239–265
 | series = Pure and Applied Mathematics
 | title = Discrete Geometry: In honor of W. Kuperberg's 60th birthday
 | year = 2003}}.</ref><ref>{{citation
 | last1 = Paffenholz | first1 = Andreas
 | last2 = Ziegler | first2 = Günter M.| author2-link = Günter M. Ziegler
 | arxiv = math.MG/0304492
 | doi = 10.1007/s00454-004-1140-4
 | issue = 4
 | journal = Discrete & Computational Geometry
 | mr = 2096750
 | pages = 601–621
 | title = The ''E<sub>t</sub>''-construction for lattices, spheres and polytopes
 | volume = 32
 | year = 2004| s2cid = 7603863
 }}.</ref>

== Semiregular polytope==
It is one of three [[Semiregular 4-polytopes#Semiregular polytopes|semiregular 4-polytopes]] made of two or more cells which are [[Platonic solid]]s, discovered by [[Thorold Gosset]] in his 1900 paper. He called it a ''tetroctahedric'' for being made of [[tetrahedron]] and [[octahedron]] cells.<ref>Gosset, 1900</ref>

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as tC<sub>5</sub>.

== Alternate names ==
* Tetroctahedric (Thorold Gosset)
* Dispentachoron
* Rectified 5-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]])
* Rectified [[Simplex|4-simplex]]
* Fully truncated 4-simplex
* Rectified pentachoron (Acronym: rap) (Jonathan Bowers)
* Ambopentachoron (Neil Sloane & [[John Horton Conway]])
* (5,2)-[[hypersimplex]] (the convex hull of five-dimensional (0,1)-vectors with exactly two ones)

==Images ==
{{4-simplex Coxeter plane graphs|t1|150}}

{| class="wikitable" width=640
|align=center|[[Image:Rectified simplex stereographic.png|220px]]<BR>[[stereographic projection]]<BR>(centered on [[octahedron]])
|align=center|[[Image:Rectified 5-cell net.png|220px]]<BR>[[Net (polytope)]]
|-
|align=center|[[Image:Rectified 5cell-perspective-tetrahedron-first-01.gif]]
|[[Tetrahedron]]-centered perspective projection into 3D space, with nearest tetrahedron to the 4D viewpoint rendered in red, and the 4 surrounding octahedra in green. Cells lying on the far side of the polytope have been culled for clarity (although they can be discerned from the edge outlines). The rotation is only of the 3D projection image, in order to show its structure, not a rotation in 4D space.
|}

== Coordinates ==
The [[Cartesian coordinate]]s of the vertices of an origin-centered rectified 5-cell having edge length&nbsp;2 are:

{| class="wikitable collapsible collapsed"
! colspan=2|Coordinates
|-
|
:<math>\left(\sqrt{\frac{2}{5}},\   \frac{2}{\sqrt{6}},\  \frac{2}{\sqrt{3}},\  0   \right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\   \frac{2}{\sqrt{6}},\  \frac{-1}{\sqrt{3}},\ \pm1\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\   \frac{-2}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\   \frac{-2}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0   \right)</math>
|
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\  \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ 0   \right)</math>
:<math>\left(\frac{-3}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\                   0   \right)</math>
|}

More simply, the vertices of the ''rectified 5-cell'' can be positioned on a [[hyperplane]] in 5-space as permutations of (0,0,0,1,1) ''or'' (0,0,1,1,1).  These construction can be seen as positive [[orthant]] facets of the [[rectified pentacross]] or [[birectified penteract]] respectively.

== Related 4-polytopes ==
The rectified 5-cell is the [[vertex figure]] of the [[5-demicube]], and the [[edge figure]] of the uniform [[2 21 polytope|2<sub>21</sub> polytope]].

=== Compound of the rectified 5-cell and its dual ===
The convex hull the rectified 5-cell and its dual (of the same long radius) is a nonuniform polychoron composed of 30 cells: 10 [[tetrahedron|tetrahedra]], 20 [[octahedron|octahedra]] (as triangular antiprisms), and 20 vertices. Its vertex figure is a [[triangular bifrustum]].

=== Pentachoron polytopes ===
The rectified 5-cell is one of 9 [[Uniform 4-polytope]]s constructed from the [3,3,3] [[Coxeter group]].
{{Pentachoron family}}

=== Semiregular polytopes ===
The rectified 5-cell is second in a dimensional series of [[Uniform k21 polytope|semiregular polytope]]s. Each progressive [[uniform polytope]] is constructed as the [[vertex figure]] of the previous polytope. [[Thorold Gosset]] identified this series in 1900 as containing all [[regular polytope]] facets, containing all [[simplex]]es and [[orthoplex]]es ([[tetrahedron]]s and [[octahedron]]s in the case of the rectified 5-cell). The [[Coxeter symbol]] for the rectified 5-cell is 0<sub>21</sub>.
{{k 21 polytopes}}

=== Isotopic polytopes===
{{Isotopic uniform simplex polytopes}}

== Notes==
{{reflist}}

== References ==
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
* [[John Horton Conway|J.H. Conway]] and [[Michael Guy (computer scientist)|M.J.T. Guy]]: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 and 39, 1965 
* [[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''', edited 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)
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, [[Chaim Goodman-Strauss]], ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 26)

==External links==
* [http://www.polytope.de/nr03.html Rectified 5-cell] - data and images
** {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron - Model 2}}
* {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x3o3o3o - rap}}

{{Polytopes}}

[[Category:Uniform 4-polytopes]]