Tetradecadokon

Rank 13 Type Regular Notation Coxeter diagram x3o3o3o3o3o3o3o3o3o3o3o3o ( ) Schläfli symbol {3,3,3,3,3,3,3,3,3,3,3,3} Elements Doka 14 tridecahenda Henda 91 dodecadaka Daka 364 hendecaxenna Xenna 1001 decayotta Yotta 2002 enneazetta Zetta 3003 octaexa Exa 3432 heptapeta Peta 3003 hexatera Tera 2002 pentachora Cells 1001 tetrahedra Faces 364 triangles Edges 91 Vertices 14 Vertex figure Tridecahendon , edge length 1Measures (edge length 1) Circumradius ${\frac {\sqrt {91}}{14}}\approx 0.68139$ Inradius ${\frac {\sqrt {91}}{182}}\approx 0.052414$ Hypervolume ${\frac {\sqrt {7}}{398529331200}}\approx 6.6388\times 10^{-12}$ Dihedral angle $\arccos \left({\frac {1}{13}}\right)\approx 85.58827^{\circ }$ Height ${\frac {\sqrt {91}}{13}}\approx 0.73380$ Central density 1 Number of external pieces 14 Level of complexity 1 Related polytopes Army Tetradecadokon Regiment Tetradecadokon Dual Tetradecadokon Conjugate None Abstract & topological properties Flag count87178291200 Euler characteristic 2 Orientable Yes Properties Symmetry A_{13} , order 87178291200Convex Yes Nature Tame

The tetradecadokon , also commonly called the 13-simplex , is the simplest possible non-degenerate polydokon . The full symmetry version has 14 regular tridecahenda as facets, joining 3 to a dakon and 13 to a vertex, and is one of the 3 regular polydoka . It is the 13-dimensional simplex .

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

$\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {55}}{11}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {66}}{12}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {78}}{13}},\,-{\frac {\sqrt {91}}{182}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {91}}{14}}\right)$ .
Much simpler coordinates can be given in 14 dimensions , as all permutations of:

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