Decayotton

Decayotton
(OFF file)
Rank	9
Type	Regular
Space	Spherical
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	Enneazetton, edge length 1
Measures (edge length 1)
Circumradius	$\frac{3\sqrt5}{10} ≈ 0.67082$
Inradius	$\frac{\sqrt5}{30} ≈ 0.074536$
Hypervolume	$\frac{\sqrt5}{5806080} ≈ 3.8513×10^{-7}$
Diyottal angle	$\arccos\left(\frac19\right) ≈ 83.62063°$
Height	$\frac{\sqrt5}{3} ≈ 0.74536$
Central density	1
Number of external pieces	10
Level of complexity	1
Related polytopes
Army	Day
Regiment	Day
Dual	Decayotton
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A9, order 3628800
Convex	Yes
Nature	Tame

The decayotton, or day, also commonly called the 9-simplex, is the simplest possible non-degenerate polyyotton. The full symmetry version has 10 regular enneazetta as facets, joining 3 to a heptapeton peak and 9 to a vertex, and is one of the 3 regular polyyotta. It is the 9-dimensional simplex.

Vertex coordinates

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

• $\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),$
• $\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),$
• $\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),$
• $\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),$
• $\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),$
• $\left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),$
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),$
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23,\,-\frac{\sqrt5}{30}\right),$
• $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{3\sqrt5}{10}\right).$

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

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

External links

• Klitzing, Richard. "day".

• Wikipedia Contributors. "9-simplex".