The 12-simplex (also called tridecahendon or tridecahendakon ) is the simplest possible non-degenerate 12-polytope . The full symmetry version has 13 regular dodecadaka as facets , joining 3 to a ridge and 12 to a vertex, and is one of the 3 convex regular 12-polytopes . It is the 12-dimensional simplex .

12-simplex

Rank 12 Type Regular Notation Coxeter diagram x3o3o3o3o3o3o3o3o3o3o3o ( ) Schläfli symbol {3,3,3,3,3,3,3,3,3,3,3} Elements Henda 13 11-simplices Daka 78 10-simplices Xenna 286 9-simplices Yotta 715 8-simplices Zetta 1287 7-simplices Exa 1716 6-simplices Peta 1716 5-simplices Tera 1287 5-cells Cells 715 tetrahedra Faces 286 triangles Edges 78 Vertices 13 Vertex figure 11-simplex , edge length 1Measures (edge length 1) Circumradius ${\frac {\sqrt {78}}{13}}\approx 0.67937$ Inradius ${\frac {\sqrt {78}}{156}}\approx 0.056614$ Hypervolume ${\frac {\sqrt {13}}{30656102400}}\approx 1.1761\times 10^{-10}$ Dihedral angle $\arccos \left({\frac {1}{12}}\right)\approx 85.21981^{\circ }$ Height ${\frac {\sqrt {78}}{12}}\approx 0.73598$ Central density 1 Number of external pieces 13 Level of complexity 1 Related polytopes Dual 12-simplex Conjugate None Convex hull 12-simplex Abstract & topological properties Flag count6227020800 Euler characteristic 0 Orientable Yes Properties Symmetry A_{12} , order 6227020800Flag orbits 1 Convex Yes Nature Tame

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

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

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