Heptadecapedakon

Heptadecapedakon
Rank	16
Type	Regular
Space	Spherical
Notation
Coxeter diagram	x3o3o3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Pedaka	17 hexadecatedaka
Tedaka	136 pentadecatradaka
Tradaka	680 tetradecadoka
Doka	2380 tridecahenda
Henda	6188 dodecadaka
Daka	12376 hendecaxenna
Xenna	19448 decayotta
Yotta	24310 enneazetta
Zetta	24310 octaexa
Exa	19448 heptapeta
Peta	12376 hexatera
Tera	6188 pentachora
Cells	2380 tetrahedra
Faces	680 triangles
Edges	136
Vertices	17
Vertex figure	Hexadecatedakon, edge length 1
Measures (edge length 1)
Circumradius	$\frac{2\sqrt{34}}{17} \approx 0.68599$
Inradius	$\frac{\sqrt{34}}{136} \approx 0.042875$
Hypervolume	$\frac{\sqrt{17}}{5356234211328000} \approx 7.6978\times 10^{-16}$
Dihedral angle	$\arccos\left(\frac{1}{16}\right) \approx 86.41668^\circ$
Height	$\frac{\sqrt{34}}{8} \approx 0.72887$
Central density	1
Number of external pieces	17
Level of complexity	1
Related polytopes
Army	Heptadecapedakon
Regiment	Heptadecapedakon
Dual	Heptadecapedakon
Conjugate	None
Abstract & topological properties
Flag count	355687428096000
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A16, order 355687428096000
Convex	Yes
Nature	Tame

The heptadecapedakon, also commonly called the 16-simplex, is the simplest possible non-degenerate polypedakon. The full symmetry version has 17 regular hexadecatedaka as facets, joining 3 to a tradakon and 16 to a vertex, and is one of the 3 regular polypedaka. It is the 16-dimensional simplex.

Vertex coordinates

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

• $\left(\pm\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23,\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{3\sqrt5}{10},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{55}}{11},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{66}}{12},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{78}}{13},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{91}}{14},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{105}}{15},\,-\frac{\sqrt{30}}{120},\,-\frac{\sqrt{34}}{136}\right)$.
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{30}}{8},\,-\frac{\sqrt{34}}{136}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{2\sqrt{34}}{17}\right)$.

Much simpler coordinates can be given in 17 dimensions, as all permutations of:

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