11-simplex

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
11-simplex
Rank11
TypeRegular
Notation
Coxeter diagramx3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3,3,3,3}
Elements
Daka12 10-simplices
Xenna66 9-simplices
Yotta220 8-simplices
Zetta495 7-simplices
Exa792 6-simplices
Peta924 5-simplices
Tera792 pentachora
Cells495 tetrahedra
Faces220 triangles
Edges66
Vertices12
Vertex figure10-simplex, edge length 1
Measures (edge length 1)
Circumradius$\displaystyle \frac{\sqrt{66}}{12} \approx 0.67700$
Inradius$\displaystyle \frac{\sqrt{66}}{132} \approx 0.067420$
Hypervolume$\displaystyle \frac{\sqrt6}{1277337600} \approx 1.9177\times10^{-9}$
Dihedral angle$\displaystyle \arccos\left(\frac{1}{11}\right) \approx 84.78409^\circ$
Height$\displaystyle \frac{\sqrt{66}}{11} \approx 0.73855$
Central density1
Number of external pieces12
Level of complexity1
Related polytopes
Army*
Regiment*
Dual11-simplex
ConjugateNone
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA11, order 479001600
Flag orbits1
ConvexYes
NatureTame

The 11-simplex, also called the dodecadakon, is the simplest possible non-degenerate 11-polytope. The full symmetry version has 12 regular 10-simplices as facets, joining 3 to a yotton peak and 11 to a vertex, and is one of the 3 regular 11-polytopes. It is the 11-dimensional simplex.

A regular 11-simplex of edge length $\displaystyle \sqrt6$ can be inscribed in the 11-cube. The next largest simplex that can be inscribed in a hypercube is the hexadecatedakon.[1]

Vertex coordinates

The vertices of a regular 11-simplex of edge length 1, centered at the origin, are given by:

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

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.