View source for 10-simplex

{{Short description|Convex regular 10-polytope}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Regular hendecaxennon<BR>(10-simplex)
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:10-simplex_t0.svg|280px]]<BR>[[Orthogonal projection]]<BR>inside [[Petrie polygon]]
|-
|bgcolor=#e7dcc3|Type||Regular [[10-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[simplex]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3,3,3,3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]|||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|9-faces||11 [[9-simplex]][[Image:9-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|8-faces||55 [[8-simplex]][[Image:8-simplex_t0.svg|25px]]
|-9
|bgcolor=#e7dcc3|7-faces||165 [[7-simplex]][[Image:7-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|6-faces||330 [[6-simplex]][[Image:6-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||462 [[5-simplex]][[Image:5-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||462 [[5-cell]][[Image:4-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||330 [[tetrahedron]][[Image:3-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||165 [[triangle]][[Image:2-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||55
|-
|bgcolor=#e7dcc3|Vertices||11
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[9-simplex]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[hendecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]|| A<sub>10</sub> [3,3,3,3,3,3,3,3,3]
|-
|bgcolor=#e7dcc3|Dual||Self-dual polytope|[[Self-dual polytope|Self-dual]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
In [[geometry]], a 10-[[simplex]] is a self-dual [[Regular polytope|regular]] [[10-polytope]]. It has 11 [[vertex (geometry)|vertices]], 55 [[Edge (geometry)|edge]]s, 165 triangle [[Face (geometry)|faces]], 330 tetrahedral [[Cell (mathematics)|cells]], 462 [[5-cell]] 4-faces, 462 [[5-simplex]] 5-faces, 330 [[6-simplex]] 6-faces, 165 [[7-simplex]] 7-faces, 55 [[8-simplex]] 8-faces, and 11 [[9-simplex]] 9-faces. Its [[dihedral angle]] is cos<sup>−1</sup>(1/10), or approximately 84.26°.

It can also be called a '''hendecaxennon''', or '''hendeca-10-tope''', as an 11-[[facet (geometry)|facetted]] polytope in 10-dimensions. Acronym: '''ux'''{{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/ux.htm (x3o3o3o3o3o3o3o3o3o – ux)]}}

The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''hendecaxennon'' is derived from ''hendeca'' for 11 [[Facet (mathematics)|facets]] in [[Greek language|Greek]] and [[Xenna|-xenn]] (variation of ennea for nine), having 9-dimensional facets, and ''-on''.

== Coordinates ==

The [[Cartesian coordinate]]s of the vertices of an origin-centered regular 10-simplex having edge length&nbsp;2 are:

:<math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 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(\sqrt{1/55},\ \sqrt{1/45},\ 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(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/55},\ \sqrt{1/45},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/55},\ -3\sqrt{1/5},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(-\sqrt{20/11},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>

More simply, the vertices of the ''10-simplex'' can be positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,0,1). This construction is based on [[Facet (geometry)|facets]] of the [[11-orthoplex]].

== Images ==

{{A10 Coxeter plane graphs|t0|100}}

== Related polytopes ==
The [[n-skeleton|2-skeleton]] of the 10-simplex is topologically related to the [[11-cell]] [[abstract polytope|abstract regular polychoron]] which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).

== References ==
{{reflist}}
* [[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 thesis |author-link=Norman Johnson (mathematician) |first=Norman |last=Johnson |title=Uniform Polytopes |date=1991 |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|polyxenna.htm|10D uniform polytopes (polyxenna)}} x3o3o3o3o3o3o3o3o3o – ux {{sfn whitelist|CITEREFKlitzing}}

== External links ==
* {{PolyCell | urlname = glossary.html#simplex| 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:10-polytopes]]