The 10-simplex , also called the hendecaxennon , is the simplest possible non-degenerate 10-polytope . The full symmetry version has 11 regular 9-simplices as facets, joining 3 to an 7-simplex peak and 10 to a vertex, and is one of the 3 regular 1-polytopes. It is the 10-dimensional simplex .

10-simplex Rank 10 Type Regular Notation Bowers style acronym Ux Coxeter diagram x3o3o3o3o3o3o3o3o3o ( ) Schläfli symbol {3,3,3,3,3,3,3,3,3} Tapertopic notation 1^{9} Elements Xenna 11 decayotta Yotta 55 enneazetta Zetta 165 octaexa Exa 330 heptapeta Peta 462 hexatera Tera 462 pentachora Cells 330 tetrahedra Faces 165 triangles Edges 55 Vertices 11 Vertex figure Decayotton , edge length 1Measures (edge length 1) Circumradius ${\frac {\sqrt {55}}{11}}\approx 0.67420$ Inradius ${\frac {\sqrt {55}}{110}}\approx 0.067420$ Hypervolume ${\frac {\sqrt {11}}{116121600}}\approx 2.8562\times 10^{-8}$ Dixennal angle $\arccos \left({\frac {1}{10}}\right)\approx 84.26083^{\circ }$ Height ${\frac {\sqrt {55}}{10}}\approx 0.74162$ Central density 1 Number of external pieces 11 Level of complexity 1 Related polytopes Army Ux Regiment Ux Dual 10-simplex Conjugate None Abstract & topological properties Flag count39916800 Euler characteristic 0 Orientable Yes Skeleton K_{11} Properties Symmetry A_{10} , order 39916800Flag orbits 1 Convex Yes Nature Tame

Vertex coordinates
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The vertices of a regular hendecaxennon of edge length 1, centered at the origin, are given by:

$\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}},\,-{\frac {\sqrt {55}}{110}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {55}}{11}}\right)$ .
Much simpler coordinates can be given in 11 dimensions , as all permutations of:

$\left({\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right)$ .
External links
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