Tridecahendon

Tridecahendon
Rank	12
Type	Regular
Space	Spherical
Notation
Coxeter diagram	x3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3,3,3,3,3,3,3}
Elements
Henda	13 dodecadaka
Daka	78 hendecaxenna
Xenna	286 decayotta
Yotta	715 enneazetta
Zetta	1287 octaexa
Exa	1716 heptapeta
Peta	1716 hexatera
Tera	1287 pentachora
Cells	715 tetrahedra
Faces	286 triangles
Edges	78
Vertices	13
Vertex figure	Dodecadakon, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt{78}}{13} \approx 0.67937$
Inradius	$\frac{\sqrt{78}}{156} \approx 0.056614$
Hypervolume	$\frac{\sqrt{13}}{30656102400} \approx 1.1761×10^{-10}$
Dihedral angle	$\arccos\left(\frac{1}{12}\right) \approx 85.21981^\circ$
Height	$\frac{\sqrt{78}}{12} \approx 0.73598$
Central density	1
Number of external pieces	13
Level of complexity	1
Related polytopes
Army	Tridecahendon
Regiment	Tridecahendon
Dual	Tridecahendon
Conjugate	None
Abstract & topological properties
Flag count	6227020800
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A12, order 6227020800
Convex	Yes
Nature	Tame

The tridecahendon (older name tridecahendakon), also commonly called the 12-simplex, is the simplest possible non-degenerate polyhendon. The full symmetry version has 13 regular dodecadaka as facets, joining 3 to a xennon and 12 to a vertex, and is one of the 3 regular polyhenda. It is the 12-dimensional simplex.

Vertex coordinates

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

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

Much simpler coordinates can be given in 13 dimensions, as all permutations of:

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