14-simplex

Rank 14 Type Regular Notation Coxeter diagram x3o3o3o3o3o3o3o3o3o3o3o3o3o ( ) Schläfli symbol {3,3,3,3,3,3,3,3,3,3,3,3,3} Elements Tradaka 15 13-simplices Doka 105 12-simplices Henda 455 11-simplices Daka 1365 10-simplices Xenna 3003 9-simplices Yotta 5005 8-simplices Zetta 6435 7-simplices Exa 6435 6-simplices Peta 5005 hexatera Tera 3003 pentachora Cells 1365 tetrahedra Faces 455 triangles Edges 105 Vertices 15 Vertex figure 13-simplex , edge length 1Measures (edge length 1) Circumradius ${\frac {\sqrt {105}}{15}}\approx 0.68313$ Inradius ${\frac {\sqrt {105}}{210}}\approx 0.048795$ Hypervolume ${\frac {\sqrt {15}}{11158821273600}}\approx 3.4708\times 10^{-13}$ Dihedral angle $\arccos \left({\frac {1}{14}}\right)\approx 85.90396^{\circ }$ Height ${\frac {\sqrt {105}}{14}}\approx 0.73193$ Central density 1 Number of external pieces 15 Level of complexity 1 Related polytopes Army 14-simplex Regiment 14-simplex Dual 14-simplex Conjugate None Abstract & topological properties Flag count1307674368000 Euler characteristic 0 Orientable Yes Properties Symmetry A_{14} , order 1307674368000Flag orbits 1 Convex Yes Nature Tame

The 14-simplex (also called the pentadecatradakon ) is the simplest possible non-degenerate 14-polytope . The full symmetry version has 15 regular 13-simplices as facets, joining 3 to a facet and 14 to a vertex, and is regular .

The vertices of a regular 14-simplex of edge length 1, centered at the origin, are given by:

$\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {55}}{11}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {66}}{12}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {78}}{13}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {91}}{14}},\,-{\frac {\sqrt {105}}{210}}\right)$ ,
$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {105}}{15}}\right)$ .
Much simpler coordinates can be given in 15 dimensions , as all permutations of:

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