Rectified 5-simplexes

{| 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 projections in A₅ Coxeter plane |} In five-dimensional geometry, a '''rectified 5-simplex''' is a convex uniform 5-polytope, being a 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 geometry, a '''rectified 5-simplex''' is a uniform 5-polytope with 15 vertices, 60 edges, 80 triangular faces, 45 cells (30 tetrahedral, and 15 octahedral), and 12 4-faces (6 5-cell and 6 rectified 5-cells). It is also called '''0_3,1''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea}}.

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)<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 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 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 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, dividing the full group order of a subgroup order by removing one mirror at a time.{{r|Klitzing rix}}

{| class=wikitable style="width:740px" !A₅||{{CDD|node|3|node_1|3|node|3|node|3|node}}||''k''-face|| f_''k'' || f₀ || f₁||colspan=2|f₂||colspan=2|f₃||colspan=2|f₄||''k''-figure|| Notes |- align=right |A₃A₁ ||{{CDD|node|2|node_x|2|node|3|node|3|node}}|| ( ) ! f₀ |BGCOLOR="#ffe0e0"|'''15'''||8||4||12||6||8||4||2 ||{3,3}×{ } || A₅/A₃A₁ = 6!/4!/2 = 15 |- align=right |A₂A₁ ||{{CDD|node_x|2|node_1|2|node_x|2|node|3|node}}|| { } ! f₁ || 2||BGCOLOR="#ffffe0"|'''60'''||1||3||3||3||3||1 ||{3}∨( ) || A₅/A₂A₁ = 6!/3!/2 = 60 |- align=right |A₂A₂ ||{{CDD|node|3|node_1|2|node_x|2|node|3|node}}|| r{3} !rowspan=2|f₂ || 3||3||BGCOLOR="#e0ffe0"|'''20'''||BGCOLOR="#e0ffe0"|*||3||0||3||0 ||{3} || A₅/A₂A₂ = 6!/3!/3! =20 |- align=right |A₂A₁ ||{{CDD|node_x|2|node_1|3|node|2|node_x|2|node}}||{3} || 3||3||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''60'''||1||2||2||1 ||{ }×( ) || A₅/A₂A₁ = 6!/3!/2 = 60 |- align=right |A₃A₁ ||{{CDD|node|3|node_1|3|node|2|node_x|2|node}}||r{3,3} !rowspan=2|f₃ || 6||12||4||4||BGCOLOR="#e0ffff"|'''15'''||BGCOLOR="#e0ffff"|*||2||0 ||rowspan=2|{ } || A₅/A₃A₁ = 6!/4!/2 = 15 |- align=right |A₃ ||{{CDD|node_x|2|node_1|3|node|3|node|2|node_x}}|| {3,3} || 4||6||0||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''30'''||1||1 || A₅/A₃ = 6!/4! = 30 |- align=right |A₄ ||{{CDD|node|3|node_1|3|node|3|node|2|node_x}}|| r{3,3,3} !rowspan=2|f₄ || 10||30||10||20||5||5||BGCOLOR="#e0e0ff"|'''6'''||BGCOLOR="#e0e0ff"|* ||rowspan=2|( ) || A₅/A₄ = 6!/5! = 6 |- align=right |A₄ ||{{CDD|node_x|2|node_1|3|node|3|node|3|node}}|| {3,3,3} || 5||10||0||10||0||5||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''6''' || A₅/A₄ = 6!/5! = 6 |}

=== Images === {| class=wikitable width=320 align=right |+ Stereographic projection |- |320px<BR>Stereographic projection of spherical form |}

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

=== Related polytopes === The rectified 5-simplex, 0₃₁, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 1_3k series. The fifth figure is a Euclidean honeycomb, 3₃₁, and the final is a noncompact hyperbolic honeycomb, 4₃₁. 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 isotopic, with all 12 of its facets as rectified 5-cells. It has 20 vertices, 90 edges, 120 triangular faces, 60 cells (30 tetrahedral, and 30 octahedral).

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

It is also called '''0_2,2''' 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 1₂₂, {{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. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). 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, dividing the full group order of a subgroup order by removing one mirror at a time.{{r|Klitzing dot}}

{| class=wikitable style="width:800px" !A₅||{{CDD|node|3|node|3|node_1|3|node|3|node}}||''k''-face|| f_''k'' || f₀ || f₁||colspan=2|f₂||colspan=3|f₃||colspan=2|f₄|| ''k''-figure|| Notes |- align=right |A₂A₂ ||{{CDD|node|3|node|2|node_x|2|node|3|node}}|| ( ) ! f₀ |BGCOLOR="#ffe0e0"|'''20'''||9||9||9||3||9||3||3||3||{3}×{3} || A₅/A₂A₂ = 6!/3!/3! = 20 |- align=right |A₁A₁A₁ ||{{CDD|node|2|node_x|2|node_1|2|node_x|2|node}}|| { } ! f₁ ||2||BGCOLOR="#ffffe0"|'''90'''||2||2||1||4||1||2||2||{ }∨{ } ||A₅/A₁A₁A₁ = 6!/2/2/2 = 90 |- align=right |A₂A₁ ||{{CDD|node_x|2|node|3|node_1|2|node_x|2|node}}||rowspan=2|{3} !rowspan=2|f₂ ||3||3||BGCOLOR="#e0ffe0"|'''60'''||BGCOLOR="#e0ffe0"|*||1||2||0||2||1||rowspan=2|{ }∨( ) ||rowspan=2| A₅/A₂A₁ = 6!/3!/2 = 60 |- align=right |A₂A₁ ||{{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 |A₃A₁ ||{{CDD|node|3|node|3|node_1|2|node_x|2|node}}|| {3,3} !rowspan=3|f₃ ||4||6||4||0||BGCOLOR="#e0ffff"|'''15'''||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||2||0||rowspan=3|{ } || A₅/A₃A₁ = 6!/4!/2 = 15 |- align=right |A₃ ||{{CDD|node_x|2|node|3|node_1|3|node|2|node_x}}|| r{3,3} ||6||12||4||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''30'''||BGCOLOR="#e0ffff"|*||1||1|| A₅/A₃ = 6!/4! = 30 |- align=right |A₃A₁ ||{{CDD|node|2|node_x|2|node_1|3|node|3|node}}|| {3,3} ||4||6||0||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''15'''||0||2|| A₅/A₃A₁ = 6!/4!/2 = 15 |- align=right |A₄ ||{{CDD|node|3|node|3|node_1|3|node|2|node_x}}|| rowspan=2|r{3,3,3} !rowspan=2|f₄ ||10||30||20||10||5||5||0||BGCOLOR="#e0e0ff"|'''6'''||BGCOLOR="#e0e0ff"|*||rowspan=2|( ) || rowspan=2|A₅/A₄ = 6!/5! = 6 |- align=right |A₄ ||{{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 |- |320px |} The ''birectified 5-simplex'' is the intersection of two regular 5-simplexes in 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 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 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'', 0₂₂, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k₂₂ series. The ''birectified 5-simplex'' is the vertex figure for the third, the 1₂₂. The fourth figure is a Euclidean honeycomb, 2₂₂, and the final is a noncompact hyperbolic honeycomb, 3₂₂. Each progressive uniform polytope is constructed from the previous as its vertex figure. {{k 22 polytopes}}

==== Isotopic 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 2₃₁ polytope.

It is also one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A₅ Coxeter plane orthographic projections. (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: ** 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 ''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