Rank14
TypeRegular
SpaceSpherical
Notation
Coxeter diagramx3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Doka105 tridecahenda
Daka1365 hendecaxenna
Xenna3003 decayotta
Yotta5005 enneazetta
Zetta6435 octaexa
Exa6435 heptapeta
Peta5005 hexatera
Tera3003 pentachora
Cells1365 tetrahedra
Faces455 triangles
Edges105
Vertices15
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{105}}{15} \approx 0.68313}$
Inradius${\displaystyle \frac{\sqrt{105}}{210} \approx 0.048795}$
Hypervolume${\displaystyle \frac{\sqrt{15}}{11158821273600} \approx 3.4708×10^{-13}}$
Dihedral angle${\displaystyle \arccos\left(\frac{1}{14}\right) \approx 85.90396°}$
Height${\displaystyle \frac{\sqrt{105}}{14} \approx 0.73193}$
Central density1
Number of external pieces15
Level of complexity1
Related polytopes
ConjugateNone
Abstract & topological properties
Flag count1307674368000
Euler characteristic0
OrientableYes
Properties
SymmetryA14, order 1307674368000
ConvexYes
NatureTame

The pentadecatradakon, also commonly called the 14-simplex, is the simplest possible non-degenerate polytradakon. The full symmetry version has 15 regular tetradecadoka as facets, joining 3 to a hendon and 14 to a vertex, and is one of the 3 regular polytradaka. It is the 14-dimensional simplex.

## Vertex coordinates

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

• ${\displaystyle \left(\pm\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23,\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{3\sqrt5}{10},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{55}}{11},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{66}}{12},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{78}}{13},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{91}}{14},\,-\frac{\sqrt{105}}{210}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{105}}{15}\right)}$.

Much simpler coordinates can be given in 15 dimensions, as all permutations of:

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