# Rectified 5-simplexes

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{| class=wikitable align=right width=450 style="margin-left:1em;"
|- align=center valign=top
|150px|class=skin-invert<BR>5-simplex<BR>{{CDD|node_1|3|node|3|node|3|node|3|node}}
|150px|class=skin-invert<BR>Rectified 5-simplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node}}
|150px|class=skin-invert<BR>Birectified 5-simplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node}}
|-
!colspan=3|[Orthogonal projection](/source/Orthogonal_projection)s in A<sub>5</sub> [Coxeter plane](/source/Coxeter_plane)
|}
In five-dimensional [geometry](/source/geometry), a '''rectified 5-simplex''' is a convex [uniform 5-polytope](/source/uniform_5-polytope), being a [rectification](/source/Rectification_(geometry)) of the regular [5-simplex](/source/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](/source/Five-dimensional_space) [geometry](/source/geometry), a '''rectified 5-simplex''' is a [uniform 5-polytope](/source/uniform_5-polytope) with 15 [vertices](/source/vertex_(geometry)), 60 [edge](/source/Edge_(geometry))s, 80 [triangular](/source/Triangle) [faces](/source/Face_(geometry)), 45 [cells](/source/Cell_(geometry)) (30 [tetrahedral](/source/Tetrahedron), and 15 [octahedral](/source/Octahedron)), and 12 [4-face](/source/4-face)s (6 [5-cell](/source/5-cell) and 6 [rectified 5-cell](/source/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}}.

[E. L. Elte](/source/Emanuel_Lodewijk_Elte) identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|5}}.

=== Alternate names ===
* Rectified hexateron (Acronym: rix) (Jonathan Bowers)<ref name="Klitzing rix">{{KlitzingPolytopes|../incmats/rix.htm|o3x3o3o3o - rix}}</ref>

=== Coordinates ===
The vertices of the rectified 5-simplex can be more simply positioned on a [hyperplane](/source/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](/source/rectified_6-orthoplex) or [birectified 6-cube](/source/birectified_6-cube) respectively.

=== As a configuration ===
This [configuration matrix](/source/Regular_4-polytope) 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 name=Coxeter>{{cite book |last=Coxeter |year=1973 |title=Regular Polytopes |title-link=Regular Polytopes (book) |at=Sec 1.8 Configurations}}</ref><ref name="Coxeter Complex">{{cite book| last=Coxeter | first=H.S.M. | title=Regular Complex Polytopes |url=https://books.google.pl/books?vid=ISBN9780521201254&redir_esc=y| publisher=Cambridge University Press |year=1975 |isbn=978-0-521-20125-4}}</ref>{{rp|p=117}}

The diagonal f-vector numbers are derived through the [Wythoff construction](/source/Wythoff_construction), dividing the full group order of a subgroup order by removing one mirror at a time.{{r|Klitzing rix}}

{| class=wikitable style="width:740px"
!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 ||[{3,3}×{ }](/source/tetrahedral_prism) || 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 ||[{3}∨( )](/source/triangular_pyramid) || 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}}|| [r{3}](/source/triangle)
!rowspan=2|f<sub>2</sub> 
||  3||3||BGCOLOR="#e0ffe0"|'''20'''||BGCOLOR="#e0ffe0"|*||3||0||3||0 ||[{3}](/source/triangle) || 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}}||[{3}](/source/triangle)
||  3||3||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''60'''||1||2||2||1 ||[{ }×( )](/source/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}}||[r{3,3}](/source/octahedron)
!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}}|| [{3,3}](/source/Tetrahedron)
||  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}}|| [r{3,3,3}](/source/Rectified_5-cell)
!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}}|| [{3,3,3}](/source/5-cell)
||  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](/source/Stereographic_projection)
|-
|320px<BR>[Stereographic projection](/source/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](/source/Coxeter) as 1<sub>3k</sub> series. The fifth figure is a Euclidean honeycomb, [3<sub>31</sub>](/source/3_31_honeycomb), and the final is a noncompact hyperbolic honeycomb, 4<sub>31</sub>. Each progressive [uniform polytope](/source/uniform_polytope) is constructed from the previous as its [vertex figure](/source/vertex_figure).
{{k 31 polytopes}}

== Birectified 5-simplex ==
{{Uniform polyteron db|Uniform polyteron stat table|dot}}
The '''birectified 5-simplex''' is [isotopic](/source/Facet-transitive), with all 12 of its facets as [rectified 5-cell](/source/rectified_5-cell)s. It has 20 [vertices](/source/vertex_(geometry)), 90 [edge](/source/Edge_(geometry))s, 120 [triangular](/source/Triangle) [faces](/source/Face_(geometry)), 60 [cells](/source/Cell_(geometry)) (30 [tetrahedral](/source/Tetrahedron), and 30 [octahedral](/source/Octahedron)).

[E. L. Elte](/source/Emanuel_Lodewijk_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](/source/vertex_figure) of the 6-dimensional [1<sub>22</sub>](/source/1_22_polytope), {{CDD|node_1|3|node|split1|nodes|3ab|nodes}}.

=== Alternate names ===
* Birectified hexateron
* dodecateron (Acronym: dot) (For 12-facetted polyteron) (Jonathan Bowers)<ref name="Klitzing dot">{{KlitzingPolytopes|../incmats/dot.htm|o3o3x3o3o - dot}}</ref>

=== Construction ===
The elements of the regular polytopes can be expressed in a [configuration matrix](/source/Configuration_(polytope)). Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts ([f-vector](/source/f-vector)s). The nondiagonal elements represent the number of row elements are incident to the column element.{{r|Coxeter|Coxeter Complex|p2=117}}
The diagonal f-vector numbers are derived through the [Wythoff construction](/source/Wythoff_construction), dividing the full group order of a subgroup order by removing one mirror at a time.{{r|Klitzing dot}}

{| class=wikitable style="width:800px"
!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}](/source/3-3_duoprism) || 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||[{ }∨{ }](/source/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|[{3}](/source/triangle)
!rowspan=2|f<sub>2</sub> 
||3||3||BGCOLOR="#e0ffe0"|'''60'''||BGCOLOR="#e0ffe0"|*||1||2||0||2||1||rowspan=2|[{ }∨( )](/source/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}}|| [{3,3}](/source/tetrahedron)
!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}}|| [r{3,3}](/source/octahedron)
||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}}|| [{3,3}](/source/tetrahedron)
||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|[r{3,3,3}](/source/rectified_5-cell)
!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](/source/Stereographic_projection)
|-
|320px
|}
The ''birectified 5-simplex'' is the [intersection](/source/intersection_(set_theory)) of two regular [5-simplex](/source/5-simplex)es in [dual](/source/dual_polytope) configuration. The vertices of a [birectification](/source/birectification) exist at the center of the faces of the original polytope(s). This intersection is analogous to the 3D [stellated octahedron](/source/stellated_octahedron), seen as a compound of two regular [tetrahedra](/source/tetrahedra) and intersected in a central [octahedron](/source/octahedron), while that is a first [rectification](/source/rectification_(geometry)) where vertices are at the center of the original edges.
{| class=wikitable width=320
|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](/source/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](/source/octahedron), and [bitruncated 5-cell](/source/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](/source/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](/source/birectified_6-orthoplex).

=== Related polytopes ===

==== ''k''<sub>22</sub> polytopes ====
The ''birectified 5-simplex'', 0<sub>22</sub>, is second in a dimensional series of uniform polytopes, expressed by [Coxeter](/source/Coxeter) as k<sub>22</sub> series. The ''birectified 5-simplex'' is the vertex figure for the third, the [1<sub>22</sub>](/source/1_22_polytope). The fourth figure is a Euclidean honeycomb, [2<sub>22</sub>](/source/2_22_honeycomb), and the final is a noncompact hyperbolic honeycomb, 3<sub>22</sub>. Each progressive [uniform polytope](/source/uniform_polytope) is constructed from the previous as its [vertex figure](/source/vertex_figure).
{{k 22 polytopes}}

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

== Related uniform 5-polytopes ==
This polytope is the [vertex figure](/source/vertex_figure) of the [6-demicube](/source/6-demicube), and the [edge figure](/source/edge_figure) of the uniform [2<sub>31</sub> polytope](/source/2_31_polytope).

It is also one of 19 [uniform polytera](/source/Uniform_polyteron) based on the [3,3,3,3] [Coxeter group](/source/Coxeter_group), all shown here in A<sub>5</sub> [Coxeter plane](/source/Coxeter_plane) [orthographic projection](/source/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}}
* [H.S.M. Coxeter](/source/Harold_Scott_MacDonald_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 Ivić Weiss, Wiley-Interscience Publication, 1995, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}
*** (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](/source/Norman_Johnson_(mathematician)) ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 
* {{KlitzingPolytopes|polytera.htm|5D uniform polytopes (polytera) with acronyms}} 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

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Adapted from the Wikipedia article [Rectified 5-simplexes](https://en.wikipedia.org/wiki/Rectified_5-simplexes) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rectified_5-simplexes?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.