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10-simplex

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Convex regular 10-polytope

Regular hendecaxennon (10-simplex)
Orthogonal projection insidePetrie polygon
Type	Regular10-polytope
Family	simplex
Schläfli symbol	{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
9-faces	119-simplex
8-faces	558-simplex
7-faces	1657-simplex
6-faces	3306-simplex
5-faces	4625-simplex
4-faces	4625-cell
Cells	330tetrahedron
Faces	165triangle
Edges	55
Vertices	11
Vertex figure	9-simplex
Petrie polygon	hendecagon
Coxeter group	A₁₀ [3,3,3,3,3,3,3,3,3]
Dual	Self-dual
Properties	convex

Ingeometry, a 10-simplex is a self-dualregular 10-polytope. It has 11vertices, 55edges, 165 trianglefaces, 330 tetrahedralcells, 4625-cell 4-faces, 4625-simplex 5-faces, 3306-simplex 6-faces, 1657-simplex 7-faces, 558-simplex 8-faces, and 119-simplex 9-faces. Itsdihedral angle is cos⁻¹(1/10), or approximately 84.26°.

It can also be called ahendecaxennon, orhendeca-10-tope, as an 11-facetted polytope in 10-dimensions. Acronym:ux^[1]

Thenamehendecaxennon is derived fromhendeca for 11facets inGreek and-xenn (variation of ennea for nine), having 9-dimensional facets, and-on.

Coordinates

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TheCartesian 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}},\ \pm 1\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 the10-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 onfacets of the11-orthoplex.

Images

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orthographic projections
A_kCoxeter plane	A₁₀	A₉	A₈
Graph
Dihedral symmetry	[11]	[10]	[9]
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Related polytopes

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The2-skeleton of the 10-simplex is topologically related to the11-cell abstract regular polychoron which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).

References

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^Klitzing,(x3o3o3o3o3o3o3o3o3o – ux).

Coxeter, H.S.M.:
- — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)".Regular Polytopes (3rd ed.). Dover. pp. 296.ISBN 0-486-61480-8.
- Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995).Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley.ISBN 978-0-471-01003-6.
  - (Paper 22)— (1940)."Regular and Semi-Regular Polytopes I".Math. Zeit.46:380–407.doi:10.1007/BF01181449.S2CID 186237114.
  - (Paper 23)— (1985)."Regular and Semi-Regular Polytopes II".Math. Zeit.188 (4):559–591.doi:10.1007/BF01161657.S2CID 120429557.
  - (Paper 24)— (1988)."Regular and Semi-Regular Polytopes III".Math. Zeit.200:3–45.doi:10.1007/BF01161745.S2CID 186237142.
Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "26. Hemicubes: 1_n1".The Symmetries of Things. p. 409.ISBN 978-1-56881-220-5.
Johnson, Norman (1991).Uniform Polytopes (Manuscript).
- Johnson, N.W. (1966).The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto.OCLC 258527038.
Klitzing, Richard."10D uniform polytopes (polyxenna)". x3o3o3o3o3o3o3o3o3o – ux

External links

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Glossary for hyperspace, George Olshevsky.
Polytopes of Various Dimensions
Multi-dimensional Glossary

v t e Fundamental convexregular anduniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) /D_n	E₆ /E₇ /E₈ /F₄ /G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron •Cube	Demicube		Dodecahedron •Icosahedron
Uniform polychoron	Pentachoron	16-cell •Tesseract	Demitesseract	24-cell	120-cell •600-cell
Uniform 5-polytope	5-simplex	5-orthoplex •5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex •6-cube	6-demicube	1₂₂ •2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex •7-cube	7-demicube	1₃₂ •2₃₁ •3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex •8-cube	8-demicube	1₄₂ •2₄₁ •4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex •9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex •10-cube	10-demicube
Uniformn-polytope	n-simplex	n-orthoplex •n-cube	n-demicube	1_k2 •2_k1 •k₂₁	n-pentagonal polytope
Topics:Polytope families •Regular polytope •List of regular polytopes and compounds •Polytope operations

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