# 12-simplex

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12-simplex
Rank12
TypeRegular
Notation
Coxeter diagramx3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3,3,3,3,3}
Elements
Henda13 11-simplices
Daka78 10-simplices
Xenna286 9-simplices
Yotta715 8-simplices
Zetta1287 7-simplices
Exa1716 6-simplices
Peta1716 5-simplices
Tera1287 5-cells
Cells715 tetrahedra
Faces286 triangles
Edges78
Vertices13
Vertex figure11-simplex, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {78}}{13}}\approx 0.67937}$
Inradius${\displaystyle {\frac {\sqrt {78}}{156}}\approx 0.056614}$
Hypervolume${\displaystyle {\frac {\sqrt {13}}{30656102400}}\approx 1.1761\times 10^{-10}}$
Dihedral angle${\displaystyle \arccos \left({\frac {1}{12}}\right)\approx 85.21981^{\circ }}$
Height${\displaystyle {\frac {\sqrt {78}}{12}}\approx 0.73598}$
Central density1
Number of external pieces13
Level of complexity1
Related polytopes
Dual12-simplex
ConjugateNone
Convex hull12-simplex
Abstract & topological properties
Flag count6227020800
Euler characteristic0
OrientableYes
Properties
SymmetryA12, order 6227020800
Flag orbits1
ConvexYes
NatureTame

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.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {55}}{11}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {66}}{12}},\,-{\frac {\sqrt {78}}{156}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {78}}{13}}\right)}$.

Much simpler coordinates can be given in 13 dimensions, as all permutations of:

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