9-simplex

9-simplex
(OFF file)
Rank	9
Type	Regular
Notation
Bowers style acronym	Day
Coxeter diagram	x3o3o3o3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3,3,3,3}
Tapertopic notation	18
Elements
Yotta	10 enneazetta
Zetta	45 octaexa
Exa	120 heptapeta
Peta	210 hexatera
Tera	252 pentachora
Cells	210 tetrahedra
Faces	120 triangles
Edges	45
Vertices	10
Vertex figure	8-simplex, edge length 1
Measures (edge length 1)
Circumradius	${\frac {3{\sqrt {5}}}{10}}\approx 0.67082$
Inradius	${\frac {\sqrt {5}}{30}}\approx 0.074536$
Hypervolume	${\frac {\sqrt {5}}{5806080}}\approx 3.8513\times 10^{-7}$
Diyottal angle	$\arccos \left({\frac {1}{9}}\right)\approx 83.62063^{\circ }$
Height	${\frac {\sqrt {5}}{3}}\approx 0.74536$
Central density	1
Number of external pieces	10
Level of complexity	1
Related polytopes
Army	Day
Regiment	Day
Dual	9-simplex
Conjugate	None
Abstract & topological properties
Flag count	3628800
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A9, order 3628800
Flag orbits	1
Convex	Yes
Nature	Tame

The 9-simplex, also called the decayotton, or day, is the simplest possible non-degenerate 9-polytope. The full symmetry version has 10 regular 8-simplices as facets, joining 3 to a 6-simplex peak and 9 to a vertex, and is one of the 3 regular 9-polytopes. It is the 9-dimensional simplex.

Vertex coordinates

The vertices of a regular 9-simplex of edge length 1, centered at the origin, are given by:

• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)$,
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)$,
• $\left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)$,
• $\left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)$,
• $\left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}}\right)$,
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}}\right)$.

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

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

External links

• Klitzing, Richard. "day".
• Wikipedia contributors. "9-simplex".