11-simplex

Rank 11 Type Regular Notation Coxeter diagram x3o3o3o3o3o3o3o3o3o3o ( ) Schläfli symbol {3,3,3,3,3,3,3,3,3,3} Elements Daka 12 10-simplices Xenna 66 9-simplices Yotta 220 8-simplices Zetta 495 7-simplices Exa 792 6-simplices Peta 924 5-simplices Tera 792 pentachora Cells 495 tetrahedra Faces 220 triangles Edges 66 Vertices 12 Vertex figure 10-simplex , edge length 1Measures (edge length 1) Circumradius ${\frac {\sqrt {66}}{12}}\approx 0.67700$ Inradius ${\frac {\sqrt {66}}{132}}\approx 0.067420$ Hypervolume ${\frac {\sqrt {6}}{1277337600}}\approx 1.9177\times 10^{-9}$ Dihedral angle $\arccos \left({\frac {1}{11}}\right)\approx 84.78409^{\circ }$ Height ${\frac {\sqrt {66}}{11}}\approx 0.73855$ Central density 1 Number of external pieces 12 Level of complexity 1 Related polytopes Army * Regiment * Dual 11-simplex Conjugate None Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry A_{11} , order 479001600Flag orbits 1 Convex Yes Nature Tame

The 11-simplex , also called the dodecadakon , is the simplest possible non-degenerate 11-polytope . The full symmetry version has 12 regular 10-simplices as facets, joining 3 to a yotton peak and 11 to a vertex, and is one of the 3 regular 11-polytopes. It is the 11-dimensional simplex .

A regular 11-simplex of edge length ${\sqrt {6}}$ can be inscribed in the 11-cube . The next largest simplex that can be inscribed in a hypercube is the hexadecatedakon .^{[1]}

The vertices of a regular 11-simplex of edge length 1, centered at the origin, are given by:

$\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {55}}{11}},\,-{\frac {\sqrt {66}}{132}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {66}}{12}}\right)$ .
Much simpler coordinates can be given in 12 dimensions , as all permutations of:

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