Rank11
TypeRegular
SpaceSpherical
Notation
Coxeter diagramx3o3o3o3o3o3o3o3o3o3o
Schläfli symbol{3,3,3,3,3,3,3,3,3,3}
Elements
Daka12 hendecaxenna
Xenna66 decayotta
Yotta220 enneazetta
Zetta495 octaexa
Exa792 heptapeta
Peta924 hexatera
Tera792 pentachora
Cells495 tetrahedra
Faces220 triangles
Edges66
Vertices12
Vertex figureHendecaxennon, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{66}}{12} ≈ 0.67700}$
Inradius${\displaystyle \frac{\sqrt{66}}{132} ≈ 0.067420}$
Hypervolume${\displaystyle \frac{\sqrt6}{1277337600} ≈ 1.9177×10^{-9}}$
Dihedral angle${\displaystyle \arccos\left(\frac{1}{11}\right) ≈ 84.78409°}$
Height${\displaystyle \frac{\sqrt{66}}{11} ≈ 0.73855}$
Central density1
Number of external pieces12
Level of complexity1
Related polytopes
Army*
Regiment*
ConjugateNone
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA11, order 479001600
ConvexYes
NatureTame

The dodecadakon, also commonly called the 11-simplex, is the simplest possible non-degenerate polydakon. The full symmetry version has 12 regular hendecaxenna as facets, joining 3 to a yotton and 11 to a vertex, and is one of the 3 regular polydaka. It is the 11-dimensional simplex.

A regular dodecadakon of edge length ${\displaystyle \sqrt6}$ can be inscribed in the hendekeract. The next largest simplex that can be inscribed in a hypercube is the hexadecatedakon.[1]

## Vertex coordinates

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

• ${\displaystyle \left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23,\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{3\sqrt5}{10},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{55}}{11},\,-\frac{\sqrt{66}}{132}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{66}}{12}\right).}$

Much simpler coordinates can be given in 12 dimensions, as all permutations of:

• ${\displaystyle \left(\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right).}$

## References

1. Sloane, N. J. A. (ed.). "Sequence A019442". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.