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10-simplex

{{Short description|Convex regular 10-polytope}}

class="wikitable" align="right" style="margin-left:10px" width="250"

!bgcolor=#e7dcc3 colspan=2|Regular hendecaxennon
(10-simplex)

bgcolor=#ffffff align=center colspan=2|280px
Orthogonal projection
inside Petrie polygon
bgcolor=#e7dcc3|TypeRegular 10-polytope
bgcolor=#e7dcc3|Familysimplex
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-faces11 9-simplex25px
bgcolor=#e7dcc3|8-faces55 8-simplex25px
9

|bgcolor=#e7dcc3|7-faces

165 7-simplex25px
bgcolor=#e7dcc3|6-faces330 6-simplex25px
bgcolor=#e7dcc3|5-faces462 5-simplex25px
bgcolor=#e7dcc3|4-faces462 5-cell25px
bgcolor=#e7dcc3|Cells330 tetrahedron25px
bgcolor=#e7dcc3|Faces165 triangle25px
bgcolor=#e7dcc3|Edges55
bgcolor=#e7dcc3|Vertices11
bgcolor=#e7dcc3|Vertex figure9-simplex
bgcolor=#e7dcc3|Petrie polygonhendecagon
bgcolor=#e7dcc3|Coxeter groupA10 [3,3,3,3,3,3,3,3,3]
bgcolor=#e7dcc3|DualSelf-dual polytope|Self-dual
bgcolor=#e7dcc3|Propertiesconvex

In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral 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−1(1/10), or approximately 84.26°.

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

The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn (variation of ennea for nine), having 9-dimensional facets, and -on.

Coordinates

The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are:

:\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)

:\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)

:\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)

:\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)

:\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)

:\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)

:\left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

:\left(\sqrt{1/55},\ \sqrt{1/45},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

:\left(\sqrt{1/55},\ -3\sqrt{1/5},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

:\left(-\sqrt{20/11},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

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 facets of the 11-orthoplex.

Images

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

Related polytopes

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

References

{{reflist}}

  • 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: 1n1 |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