Rectified 8-simplexes

{| class="wikitable skin-invert-image" style="float:right; margin-left:1em" |- align=center valign=top |150px 8-simplex |150px Rectified 8-simplex |- align=center valign=top |150px Birectified 8-simplex |150px Trirectified 8-simplex |- !colspan=4|Orthogonal projections in A8 Coxeter plane |} In eight-dimensional geometry, a '''rectified 8-simplex''' is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex. {{clear}}

== Rectified 8-simplex == {| class="wikitable" style="float:right; margin-left:10px; width:250px" ! style="background:#e7dcc3;" colspan="2"|Rectified 8-simplex |- | style="background:#e7dcc3;"|Type||uniform 8-polytope |- |bgcolor=#e7dcc3|Coxeter symbol|| 061 |- | style="background:#e7dcc3;"|Schläfli symbol|| t1{37} r{37} = {36,1} or <math>\left\{\begin{array}{l}3, 3, 3, 3, 3,3\\3\end{array}\right\}</math> |- | style="background:#e7dcc3;"|Coxeter-Dynkin diagrams||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}} or {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}} |- | style="background:#e7dcc3;"|7-faces||18 |- | style="background:#e7dcc3;"|6-faces||108 |- | style="background:#e7dcc3;"|5-faces||336 |- | style="background:#e7dcc3;"|4-faces||630 |- | style="background:#e7dcc3;"|Cells||756 |- | style="background:#e7dcc3;"|Faces||588 |- | style="background:#e7dcc3;"|Edges||252 |- | style="background:#e7dcc3;"|Vertices||36 |- | style="background:#e7dcc3;"|Vertex figure||7-simplex prism, {}×{3,3,3,3,3} |- | style="background:#e7dcc3;"|Petrie polygon||enneagon |- | style="background:#e7dcc3;"|Coxeter group||A8, [37], order 362880 |- | style="background:#e7dcc3;"|Properties||convex |} E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|8}}. It is also called '''06,1''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}. Acronym: rene (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/rene.htm (o3x3o3o3o3o3o3o - rene)]}}

The rectified 8-simplex is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb.

=== Coordinates === The Cartesian coordinates of the vertices of the ''rectified 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 9-orthoplex.

=== Images === {{8-simplex Coxeter plane graphs|t1|120}} {{-}}

== Birectified 8-simplex == {| class="wikitable" style="float:right; margin-left:10px; width:280px" ! style="background:#e7dcc3;" colspan="2"|Birectified 8-simplex |- | style="background:#e7dcc3;"|Type||uniform 8-polytope |- |bgcolor=#e7dcc3|Coxeter symbol|| 052 |- | style="background:#e7dcc3;"|Schläfli symbol|| t2{37} 2r{37} = {35,2} or <math>\left\{\begin{array}{l}3, 3, 3, 3, 3\\3, 3\end{array}\right\}</math> |- | style="background:#e7dcc3;"|Coxeter-Dynkin diagrams||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}} or {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea|3a|nodea}} |- | style="background:#e7dcc3;"|7-faces||18 |- | style="background:#e7dcc3;"|6-faces||144 |- | style="background:#e7dcc3;"|5-faces||588 |- | style="background:#e7dcc3;"|4-faces||1386 |- | style="background:#e7dcc3;"|Cells||2016 |- | style="background:#e7dcc3;"|Faces||1764 |- | style="background:#e7dcc3;"|Edges||756 |- | style="background:#e7dcc3;"|Vertices||84 |- | style="background:#e7dcc3;"|Vertex figure||{3}×{3,3,3,3} |- | style="background:#e7dcc3;"|Coxeter group||A8, [37], order 362880 |- | style="background:#e7dcc3;"|Properties||convex |} E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|2|8}}. It is also called '''05,2''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea|3a|nodea}}. Acronym: brene (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/brene.htm (o3o3x3o3o3o3o3o - brene)]}}

The birectified 8-simplex is the vertex figure of the 152 honeycomb.

=== Coordinates === The Cartesian coordinates of the vertices of the ''birectified 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 9-orthoplex.

=== Images === {{8-simplex Coxeter plane graphs|t2|120}}

== Trirectified 8-simplex == {| class="wikitable" style="float:right; margin-left:10px; width:280px" ! style="background:#e7dcc3;" colspan="2"|Trirectified 8-simplex |- | style="background:#e7dcc3;"|Type||uniform 8-polytope |- |bgcolor=#e7dcc3|Coxeter symbol|| 043 |- | style="background:#e7dcc3;"|Schläfli symbol|| t3{37} 3r{37} = {34,3} or <math>\left\{\begin{array}{l}3, 3, 3, 3\\3, 3,3 \end{array}\right\}</math> |- | style="background:#e7dcc3;"|Coxeter-Dynkin diagrams||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}} or {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3a|nodea}} |- | style="background:#e7dcc3;"|7-faces||9 + 9 |- | style="background:#e7dcc3;"|6-faces|| 36 + 72 + 36 |- | style="background:#e7dcc3;"|5-faces|| 84 + 252 + 252 + 84 |- | style="background:#e7dcc3;"|4-faces|| 126 + 504 + 756 + 504 |- | style="background:#e7dcc3;"|Cells|| 630 + 1260 + 1260 |- | style="background:#e7dcc3;"|Faces|| 1260 + 1680 |- | style="background:#e7dcc3;"|Edges||1260 |- | style="background:#e7dcc3;"|Vertices||126 |- | style="background:#e7dcc3;"|Vertex figure||{3,3}×{3,3,3} |- | style="background:#e7dcc3;"|Petrie polygon||enneagon |- | style="background:#e7dcc3;"|Coxeter group||A7, [37], order 362880 |- | style="background:#e7dcc3;"|Properties||convex |} E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|3|8}}. It is also called '''04,3''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3a|nodea}}. Acronym: trene (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/trene.htm (o3o3o3x3o3o3o3o - trene)]}}

=== Coordinates === The Cartesian coordinates of the vertices of the ''trirectified 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 9-orthoplex.

=== Images === {{8-simplex Coxeter plane graphs|t3|120}}

== Related polytopes == The three presented polytopes are in the family of 135 uniform 8-polytopes with A8 symmetry. {{Enneazetton family}}

== Notes == {{reflist}}

== References == * 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|polyzetta.htm|8D uniform polytopes (polyzetta) with acronyms}} o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene {{sfn whitelist| CITEREFKlitzing}}

== External links == * [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions] * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary] {{Polytopes}}

Category:8-polytopes