Pentadecatradakon

Pentadecatradakon
Rank	14
Type	Regular
Space	Spherical
Notation
Coxeter diagram	x3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Tradaka	15 tetradecadoka
Doka	105 tridecahenda
Henda	455 dodecadaka
Daka	1365 hendecaxenna
Xenna	3003 decayotta
Yotta	5005 enneazetta
Zetta	6435 octaexa
Exa	6435 heptapeta
Peta	5005 hexatera
Tera	3003 pentachora
Cells	1365 tetrahedra
Faces	455 triangles
Edges	105
Vertices	15
Vertex figure	Tetradecadokon, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt{105}}{15} \approx 0.68313$
Inradius	$\frac{\sqrt{105}}{210} \approx 0.048795$
Hypervolume	$\frac{\sqrt{15}}{11158821273600} \approx 3.4708×10^{-13}$
Dihedral angle	$\arccos\left(\frac{1}{14}\right) \approx 85.90396°$
Height	$\frac{\sqrt{105}}{14} \approx 0.73193$
Central density	1
Number of external pieces	15
Level of complexity	1
Related polytopes
Army	Pentadecatradakon
Regiment	Pentadecatradakon
Dual	Pentadecatradakon
Conjugate	None
Abstract & topological properties
Flag count	1307674368000
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A14, order 1307674368000
Convex	Yes
Nature	Tame

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:

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

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

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