View source for 8-simplex

{{Short description|Convex regular 8-polytope}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Regular enneazetton<br />(8-simplex)
|-
|  style="background:#fff; text-align:center;" colspan="2"|[[File:8-simplex t0.svg|280px]]<br />[[Orthogonal projection]]<br />inside [[Petrie polygon]]
|-
| style="background:#e7dcc3;"|Type||Regular [[8-polytope]]
|-
| style="background:#e7dcc3;"|Family||[[simplex]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| {3,3,3,3,3,3,3}
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]||{{CDD||node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
| style="background:#e7dcc3;"|7-faces||9 [[7-simplex]] [[File:7-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|6-faces||36 [[6-simplex]] [[File:6-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|5-faces||84 [[5-simplex]] [[File:5-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|4-faces||126 [[5-cell]] [[File:4-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|Cells||126 [[tetrahedron]][[File:3-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|Faces||84 [[triangle]][[File:2-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|Edges||36
|-
| style="background:#e7dcc3;"|Vertices||9
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||[[7-simplex]]
|-
| style="background:#e7dcc3;"|[[Petrie polygon]]||[[enneagon]]
|-
| style="background:#e7dcc3;"|[[Coxeter group]]|| A<sub>8</sub> [3,3,3,3,3,3,3]
|-
| style="background:#e7dcc3;"|Dual||[[Self-dual polytope|Self-dual]]
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}
In [[geometry]], an 8-[[simplex]] is a self-dual [[Regular polytope|regular]] [[8-polytope]]. It has 9 [[vertex (geometry)|vertices]], 36 [[Edge (geometry)|edges]], 84 triangle [[Face (geometry)|faces]], 126 tetrahedral [[Cell (mathematics)|cells]], 126 [[5-cell]] 4-faces, 84 [[5-simplex]] 5-faces, 36 [[6-simplex]] 6-faces, and 9 [[7-simplex]] 7-faces. Its [[dihedral angle]] is cos<sup>−1</sup>(1/8), or approximately 82.82°.

It can also be called an '''enneazetton''', or '''ennea-8-tope''', as a 9-[[facet (geometry)|facetted]] polytope in eight-dimensions. The [[5-polytope#A note on generalitay of terms for n-polytopes and elements|name]] ''enneazetton'' is derived from ''ennea'' for nine [[Facet (mathematics)|facets]] in [[Greek language|Greek]] and [[Zetta|''-zetta'']] for having seven-dimensional facets, with suffix ''-on''.<br />
Jonathan Bowers gives it the acronym '''ene'''.<ref>{{harvnb|Klitzing|at=[https://bendwavy.org/klitzing/incmats/ene.htm (x3o3o3o3o3o3o3o – ene)]}}</ref>

== As a configuration ==
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.<ref>{{harvnb|Coxeter|1973|loc=§1.8 Configurations}}</ref><ref>{{cite book |first=H.S.M. |last=Coxeter |title=Regular Complex Polytopes |url=https://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA117 |pages=117 |edition=2nd |publisher=Cambridge University Press |year=1991 |isbn=9780521394901}}</ref>

<math>\begin{bmatrix}\begin{matrix}
9 &  8 & 28 &  56 &  70 & 56 & 28 & 8
\\ 2 & 36 &  7 &  21 &  35 & 35 & 21 & 7
\\ 3 &  3 & 84 &   6 &  15 & 20 & 15 & 6
\\ 4 &  6 &  4 & 126 &   5 & 10 & 10 & 5
\\ 5 & 10 & 10 &   5 & 126 &  4 &  6 & 4
\\ 6 & 15 & 20 &  15 &   6 & 84 &  3 & 3
\\ 7 & 21 & 35 &  35 &  21 &  7 & 36 & 2
\\ 8 & 28 & 56 &  70 &  56 & 28 &  8 & 9
\end{matrix}\end{bmatrix}</math>

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

:<math>\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)</math>
:<math>\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)</math>
:<math>\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
:<math>\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
:<math>\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>

More simply, the vertices of the ''8-simplex'' can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on [[Facet (geometry)|facets]] of the [[9-orthoplex]].

Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3  and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.

== Images ==
[[File:K9-gyroelongated square pyramid.gif|thumb|The skeleton can be projected into the 9 vertices of a [[gyroelongated square pyramid]], edges colored by length.]]

{{8-simplex Coxeter plane graphs|t0|100}}

== Related polytopes and honeycombs ==
This polytope is a facet in the uniform tessellations: [[2 51 honeycomb|2<sub>51</sub>]], and [[5 21 honeycomb|5<sub>21</sub>]] with respective [[Coxeter-Dynkin diagram]]s:
:{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}, {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}

This polytope is one of 135 [[8-polytope#The A8 .5B3,3,3,3,3,3,3.5D family (8-simplex)|uniform 8-polytopes]] with A<sub>8</sub> symmetry.
{{Enneazetton family}}

== References ==
<references />
* [[Harold Scott MacDonald Coxeter|Coxeter, H.S.M.]]: 
** {{cite book |title-link=Regular Polytopes (book) |author-mask=1 |first=H.S.M. |last=Coxeter |chapter=Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) |title=Regular Polytopes |publisher=Dover |edition=3rd |year=1973 |isbn=0-486-61480-8 |pages=[https://archive.org/details/regularpolytopes00coxe_869/page/n319 296] }}
** {{cite book |editor-first=F. Arthur |editor-last=Sherk |editor2-first=Peter |editor2-last=McMullen |editor3-first=Anthony C. |editor3-last=Thompson |editor4-first=Asia Ivic |editor4-last=Weiss |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |publisher=Wiley  |year=1995 |isbn=978-0-471-01003-6 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PP1}}
*** (Paper 22) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi Regular Polytopes I |journal=Math. Zeit. |volume=46 |pages=380–407 |year=1940 |doi=10.1007/BF01181449 |s2cid=186237114 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA251 |url-access=subscription }}
*** (Paper 23) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes II |journal=Math. Zeit. |volume=188 |pages=559–591 |year=1985 |issue=4 |doi=10.1007/BF01161657 |s2cid=120429557 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA279|url-access=subscription }}
*** (Paper 24) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes III |journal=Math. Zeit. |volume=200 |pages=3–45 |year=1988 |doi=10.1007/BF01161745 |s2cid=186237142 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA313|url-access=subscription }}
* {{cite book |author-link=John Horton Conway |first1=John H. |last1=Conway |first2=Heidi |last2=Burgiel |first3=Chaim |last3=Goodman-Strauss |chapter=26. Hemicubes: 1<sub>n1</sub> |title=The Symmetries of Things |year=2008 |isbn=978-1-56881-220-5 |pages=409 }}
* {{cite document |author-link=Norman Johnson (mathematician) |first=Norman |last=Johnson |title=Uniform Polytopes |date=1991|publisher= Norman Johnson (mathematician) |type=Manuscript }}
** {{cite thesis |first=N.W. |last=Johnson |title=The Theory of Uniform Polytopes and Honeycombs |date=1966 |type=PhD |publisher=University of Toronto |url=https://search.library.utoronto.ca/details?402790 |oclc=258527038}}
* {{KlitzingPolytopes|polyzetta.htm|8D uniform polytopes (polyzetta) with acronyms}} (x3o3o3o3o3o3o3o – ene) {{sfn whitelist|CITEREFKlitzing}}

== External links ==
* {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:8-polytopes]]