View source for Rectified 5-simplexes

{| class=wikitable align=right width=450 style="margin-left:1em;"
|- align=center valign=top
|[[File:5-simplex t0.svg|150px]]<BR>5-simplex<BR>{{CDD|node_1|3|node|3|node|3|node|3|node}}
|[[File:5-simplex t1.svg|150px]]<BR>Rectified 5-simplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node}}
|[[File:5-simplex t2.svg|150px]]<BR>Birectified 5-simplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node}}
|-
!colspan=3|[[Orthogonal projection]]s in A<sub>5</sub> [[Coxeter plane]]
|}
In five-dimensional [[geometry]], a '''rectified 5-simplex''' is a convex [[uniform 5-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[5-simplex]].

There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ''rectified 5-simplex'' are located at the edge-centers of the ''5-simplex''. Vertices of the ''birectified 5-simplex'' are located in the triangular face centers of the ''5-simplex''.

==Rectified 5-simplex==
{{Uniform polyteron db|Uniform polyteron stat table|rix}}

In [[Five-dimensional space|five-dimensional]] [[geometry]], a '''rectified 5-simplex''' is a [[uniform 5-polytope]] with 15 [[vertex (geometry)|vertices]], 60 [[Edge (geometry)|edge]]s, 80 [[Triangle|triangular]] [[Face (geometry)|faces]], 45 [[Cell (geometry)|cells]] (30 [[Tetrahedron|tetrahedral]], and 15 [[Octahedron|octahedral]]), and 12 [[4-face]]s (6 [[5-cell]] and 6 [[rectified 5-cell]]s). It is also called '''0<sub>3,1</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea}}.

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|5}}.

=== Alternate names ===
* Rectified hexateron (Acronym: rix) (Jonathan Bowers)

=== Coordinates ===

The vertices of the rectified 5-simplex can be more simply positioned on a [[hyperplane]] in 6-space as permutations of (0,0,0,0,1,1) ''or'' (0,0,1,1,1,1).  These construction can be seen as facets of the [[rectified 6-orthoplex]] or [[birectified 6-cube]] respectively.

=== As a configuration ===
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the rectified 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole rectified 5-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>

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/rix.htm|o3x3o3o3o - rix}}</ref>

{| class=wikitable
!A<sub>5</sub>||{{CDD|node|3|node_1|3|node|3|node|3|node}}||k-face|| f<sub>k</sub> || f<sub>0</sub> || f<sub>1</sub>||colspan=2|f<sub>2</sub>||colspan=2|f<sub>3</sub>||colspan=2|f<sub>4</sub>|| ''k''-figure|| notes
|- align=right
|A<sub>3</sub>A<sub>1</sub> ||{{CDD|node|2|node_x|2|node|3|node|3|node}}|| ( )
! f<sub>0</sub> 
|BGCOLOR="#ffe0e0"|'''15'''||8||4||12||6||8||4||2 ||[[tetrahedral prism|{3,3}×{ }]] || A<sub>5</sub>/A<sub>3</sub>A<sub>1</sub> = 6!/4!/2 = 15
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node_x|2|node_1|2|node_x|2|node|3|node}}|| { } 
! f<sub>1</sub> 
||  2||BGCOLOR="#ffffe0"|'''60'''||1||3||3||3||3||1 ||[[triangular pyramid|{3}∨( )]] || A<sub>5</sub>/A<sub>2</sub>A<sub>1</sub> = 6!/3!/2 = 60
|- align=right
|A<sub>2</sub>A<sub>2</sub> ||{{CDD|node|3|node_1|2|node_x|2|node|3|node}}|| [[triangle|r{3}]]
!rowspan=2|f<sub>2</sub> 
||  3||3||BGCOLOR="#e0ffe0"|'''20'''||BGCOLOR="#e0ffe0"|*||3||0||3||0 ||[[triangle|{3}]] || A<sub>5</sub>/A<sub>2</sub>A<sub>2</sub> = 6!/3!/3! =20
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node_x|2|node_1|3|node|2|node_x|2|node}}||[[triangle|{3}]]
||  3||3||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''60'''||1||2||2||1 ||[[Isosceles triangle|{ }×( )]] || A<sub>5</sub>/A<sub>2</sub>A<sub>1</sub> = 6!/3!/2 = 60
|- align=right
|A<sub>3</sub>A<sub>1</sub> ||{{CDD|node|3|node_1|3|node|2|node_x|2|node}}||[[octahedron|r{3,3}]]
!rowspan=2|f<sub>3</sub> 
|| 6||12||4||4||BGCOLOR="#e0ffff"|'''15'''||BGCOLOR="#e0ffff"|*||2||0 ||rowspan=2|{ } || A<sub>5</sub>/A<sub>3</sub>A<sub>1</sub> = 6!/4!/2 = 15
|- align=right
|A<sub>3</sub> ||{{CDD|node_x|2|node_1|3|node|3|node|2|node_x}}|| [[Tetrahedron|{3,3}]]
||  4||6||0||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''30'''||1||1 || A<sub>5</sub>/A<sub>3</sub> = 6!/4! = 30
|- align=right
|A<sub>4</sub> ||{{CDD|node|3|node_1|3|node|3|node|2|node_x}}|| [[Rectified 5-cell|r{3,3,3}]]
!rowspan=2|f<sub>4</sub> 
|| 10||30||10||20||5||5||BGCOLOR="#e0e0ff"|'''6'''||BGCOLOR="#e0e0ff"|* ||rowspan=2|( ) || A<sub>5</sub>/A<sub>4</sub> = 6!/5! = 6
|- align=right
|A<sub>4</sub> ||{{CDD|node_x|2|node_1|3|node|3|node|3|node}}|| [[5-cell|{3,3,3}]]
||  5||10||0||10||0||5||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''6''' || A<sub>5</sub>/A<sub>4</sub> = 6!/5! = 6
|}

=== Images ===

{| class=wikitable width=320 align=right
|+ [[Stereographic projection]]
|-
|[[Image:Rectified Hexateron.png|320px]]<BR>[[Stereographic projection]] of spherical form
|}

{{5-simplex Coxeter plane graphs|t1|100}}

=== Related polytopes===
The rectified 5-simplex,  0<sub>31</sub>, is second in a dimensional series of uniform polytopes, expressed by [[Coxeter]] as 1<sub>3k</sub> series. The fifth figure is a Euclidean honeycomb, [[3 31 honeycomb|3<sub>31</sub>]], and the final is a noncompact hyperbolic honeycomb, 4<sub>31</sub>. Each progressive [[uniform polytope]] is constructed from the previous as its [[vertex figure]].
{{k 31 polytopes}}

==Birectified 5-simplex==
{{Uniform polyteron db|Uniform polyteron stat table|dot}}
The '''birectified 5-simplex''' is [[Facet-transitive|isotopic]], with all 12 of its facets as [[rectified 5-cell]]s. It has 20 [[vertex (geometry)|vertices]], 90 [[Edge (geometry)|edge]]s, 120 [[Triangle|triangular]] [[Face (geometry)|faces]], 60 [[Cell (geometry)|cells]] (30 [[Tetrahedron|tetrahedral]], and 30 [[Octahedron|octahedral]]).

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|2|5}}.

It is also called '''0<sub>2,2</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes}}. It is seen in the [[vertex figure]] of the 6-dimensional [[1  22 polytope|1<sub>22</sub>]], {{CDD|node_1|3|node|split1|nodes|3ab|nodes}}.

=== Alternate names ===
* Birectified hexateron
* dodecateron (Acronym: dot) (For 12-facetted polyteron) (Jonathan Bowers)

=== Construction ===
The elements of the regular polytopes can be expressed in a [[Configuration (polytope)|configuration matrix]]. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts ([[f-vector]]s). The nondiagonal elements represent the number of row elements are incident to the column element.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>

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/dot.htm|o3o3x3o3o - dot}}</ref>

{| class=wikitable
!A<sub>5</sub>||{{CDD|node|3|node|3|node_1|3|node|3|node}}||k-face|| f<sub>k</sub> || f<sub>0</sub> || f<sub>1</sub>||colspan=2|f<sub>2</sub>||colspan=3|f<sub>3</sub>||colspan=2|f<sub>4</sub>|| ''k''-figure|| notes
|- align=right
|A<sub>2</sub>A<sub>2</sub> ||{{CDD|node|3|node|2|node_x|2|node|3|node}}|| ( )
! f<sub>0</sub> 
|BGCOLOR="#ffe0e0"|'''20'''||9||9||9||3||9||3||3||3||[[3-3 duoprism|{3}×{3}]] || A<sub>5</sub>/A<sub>2</sub>A<sub>2</sub> = 6!/3!/3! = 20
|- align=right
|A<sub>1</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|node|2|node_x|2|node_1|2|node_x|2|node}}|| { }
! f<sub>1</sub> 
||2||BGCOLOR="#ffffe0"|'''90'''||2||2||1||4||1||2||2||[[tetragonal disphenoid|{ }∨{ }]] ||A<sub>5</sub>/A<sub>1</sub>A<sub>1</sub>A<sub>1</sub> = 6!/2/2/2 = 90
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node_x|2|node|3|node_1|2|node_x|2|node}}||rowspan=2|[[triangle|{3}]]
!rowspan=2|f<sub>2</sub> 
||3||3||BGCOLOR="#e0ffe0"|'''60'''||BGCOLOR="#e0ffe0"|*||1||2||0||2||1||rowspan=2|[[Isosceles triangle|{ }∨( )]] ||rowspan=2| A<sub>5</sub>/A<sub>2</sub>A<sub>1</sub> = 6!/3!/2 = 60
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node|2|node_x|2|node_1|3|node|2|node_x}}
||3||3||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''60'''||0||2||1||1||2
|- align=right
|A<sub>3</sub>A<sub>1</sub> ||{{CDD|node|3|node|3|node_1|2|node_x|2|node}}|| [[tetrahedron|{3,3}]]
!rowspan=3|f<sub>3</sub> 
||4||6||4||0||BGCOLOR="#e0ffff"|'''15'''||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||2||0||rowspan=3|{ } || A<sub>5</sub>/A<sub>3</sub>A<sub>1</sub> = 6!/4!/2 = 15
|- align=right
|A<sub>3</sub> ||{{CDD|node_x|2|node|3|node_1|3|node|2|node_x}}|| [[octahedron|r{3,3}]]
||6||12||4||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''30'''||BGCOLOR="#e0ffff"|*||1||1|| A<sub>5</sub>/A<sub>3</sub> = 6!/4! = 30
|- align=right
|A<sub>3</sub>A<sub>1</sub> ||{{CDD|node|2|node_x|2|node_1|3|node|3|node}}|| [[tetrahedron|{3,3}]]
||4||6||0||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''15'''||0||2|| A<sub>5</sub>/A<sub>3</sub>A<sub>1</sub> = 6!/4!/2 = 15
|- align=right
|A<sub>4</sub> ||{{CDD|node|3|node|3|node_1|3|node|2|node_x}}|| rowspan=2|[[rectified 5-cell|r{3,3,3}]]
!rowspan=2|f<sub>4</sub> 
||10||30||20||10||5||5||0||BGCOLOR="#e0e0ff"|'''6'''||BGCOLOR="#e0e0ff"|*||rowspan=2|( ) || rowspan=2|A<sub>5</sub>/A<sub>4</sub> = 6!/5! = 6
|- align=right
|A<sub>4</sub> ||{{CDD|node_x|2|node|3|node_1|3|node|3|node}}
||10||30||10||20||0||5||5||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''6'''
|}

=== Images ===

The A5 projection has an identical appearance to ''Metatron's Cube''.<ref>{{cite book |last= Melchizedek |first= Drunvalo |title= The Ancient Secret of the Flower of Life|publisher= Light Technology Publishing | date=1999 |volume=1 }} p.160 Figure 6-12</ref>

{{5-simplex2 Coxeter plane graphs|t2|100}}

=== Intersection of two 5-simplices ===
{| class=wikitable width=320 align=right
|+ [[Stereographic projection]]
|-
|[[Image:Birectified Hexateron.png|320px]]
|}
The ''birectified 5-simplex'' is the [[intersection (set theory)|intersection]] of two regular [[5-simplex]]es in [[dual polytope|dual]] configuration. The vertices of a [[birectification]] exist at the center of the faces of the original polytope(s). This intersection is analogous to the 3D [[stellated octahedron]], seen as a compound of two regular [[tetrahedra]] and intersected in a central [[octahedron]], while that is a first [[rectification (geometry)|rectification]] where vertices are at the center of the original edges.
{| class=wikitable width=320
|[[File:Dual_5-simplex_intersection_graphs.png|320px]]
|-
|Dual 5-simplexes (red and blue), and their birectified 5-simplex intersection in green, viewed in A5 and A4 Coxeter planes. The simplexes overlap in the A5 projection and are drawn in magenta.
|}
It is also the intersection of a [[6-cube]] with the hyperplane that bisects the 6-cube's long diagonal orthogonally.  In this sense it is the 5-dimensional analog of the regular hexagon, [[octahedron]], and [[bitruncated 5-cell]].  This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space: the 20 distinct permutations of (1,1,1,−1,−1,−1).

The vertices of the ''birectified 5-simplex'' can also be positioned on a [[hyperplane]] in 6-space as permutations of (0,0,0,1,1,1).  This construction can be seen as facets of the [[birectified 6-orthoplex]].

=== Related polytopes===

==== k_22 polytopes ====

The ''birectified 5-simplex'', 0<sub>22</sub>, is second in a dimensional series of uniform polytopes, expressed by [[Coxeter]] as k<sub>22</sub> series. The ''birectified 5-simplex'' is the vertex figure for the third, the [[1 22 polytope|1<sub>22</sub>]]. The fourth figure is a Euclidean honeycomb, [[2 22 honeycomb|2<sub>22</sub>]], and the final is a noncompact hyperbolic honeycomb, 3<sub>22</sub>. Each progressive [[uniform polytope]] is constructed from the previous as its [[vertex figure]].
{{k 22 polytopes}}

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

== Related uniform 5-polytopes ==

This polytope is the [[vertex figure]] of the [[6-demicube]], and the [[edge figure]] of the uniform [[2 31 polytope|2<sub>31</sub> polytope]].

It is also one 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)

{{Hexateron family}}

== References ==
{{reflist}}
* [[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. 
* {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} o3x3o3o3o - rix, o3o3x3o3o - dot

== External links ==
* {{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
** [http://www.polytope.net/hedrondude/rectates5.htm Rectified uniform polytera] (Rix), Jonathan Bowers
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:5-polytopes]]