Tetradecadokon

Tetradecadokon
Rank	13
Type	Regular
Space	Spherical
Notation
Coxeter diagram	x3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Doka	14 tridecahenda
Henda	91 dodecadaka
Daka	364 hendecaxenna
Xenna	1001 decayotta
Yotta	2002 enneazetta
Zetta	3003 octaexa
Exa	3432 heptapeta
Peta	3003 hexatera
Tera	2002 pentachora
Cells	1001 tetrahedra
Faces	364 triangles
Edges	91
Vertices	14
Vertex figure	Tridecahendon, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt{91}}{14} \approx 0.68139$
Inradius	$\frac{\sqrt{91}}{182} \approx 0.052414$
Hypervolume	$\frac{\sqrt7}{398529331200} \approx 6.6388×10^{-12}$
Dihedral angle	$\arccos\left(\frac{1}{13}\right) \approx 85.58827^\circ$
Height	$\frac{\sqrt{91}}{13} \approx 0.73380$
Central density	1
Number of external pieces	14
Level of complexity	1
Related polytopes
Army	Tetradecadokon
Regiment	Tetradecadokon
Dual	Tetradecadokon
Conjugate	None
Abstract & topological properties
Flag count	87178291200
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A13, order 87178291200
Convex	Yes
Nature	Tame

The tetradecadokon, also commonly called the 13-simplex, is the simplest possible non-degenerate polydokon. The full symmetry version has 14 regular tridecahenda as facets, joining 3 to a dakon and 13 to a vertex, and is one of the 3 regular polydoka. It is the 13-dimensional simplex.

Vertex coordinates

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

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

Much simpler coordinates can be given in 14 dimensions, as all permutations of:

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