Trirectified 9-simplex

Trirectified 9-simplex
Rank	9
Type	Uniform
Notation
Bowers style acronym	Treday
Coxeter diagram	o3o3o3x3o3o3o3o3o ()
Elements
Yotta	10 birectified 8-simplices 10 trirectified 8-simplices
Zetta	45 rectified 7-simplices 90 birectified 7-simplices 45 trirectified 7-simplices
Exa	120 6-simplices 360 rectified 6-simplices 120+360 birectified 6-simplices
Peta	840 5-simplices 210+1260 rectified 5-simplices 840 birectified 5-simplices
Tera	252+2520 pentachora 1260+2520 rectified pentachora
Cells	1260+4200 tetrahedra 3150 octahedra
Faces	2520+4200 triangles
Edges	2520
Vertices	210
Vertex figure	Tetrahedral-hexateric duoprism, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {30}}{5}}\approx 1.09545$
Hypervolume	${\frac {3252599{\sqrt {5}}}{29030400}}\approx 0.025053$
Diyottal angles	Trene–broc–brene: $\arccos \left(-{\frac {1}{9}}\right)\approx 96.38037^{\circ }$
	Trene–he–trene: $\arccos \left({\frac {1}{9}}\right)\approx 83.62063^{\circ }$
	Brene–roc–brene: $\arccos \left({\frac {1}{9}}\right)\approx 83.62063^{\circ }$
Height	${\frac {\sqrt {5}}{3}}\approx 0.74536$
Central density	1
Number of external pieces	20
Level of complexity	56
Related polytopes
Army	Treday
Regiment	Treday
Conjugate	None
Abstract & topological properties
Flag count	203212800
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A9, order 3628800
Flag orbits	56
Convex	Yes
Nature	Tame

The trirectified 9-simplex, also called the trirectified decayotton, or treday, is a convex uniform 9-polytope. It consists of 9 birectified 8-simplices and 9 trirectified 8-simplices. 4 birectified 8-simplices and 6 trirectified 8-simplices join at each tetrahedral-hexateric duoprismatic vertex. As the name suggests, it is the trirectification of the 9-simplex.

It is also a convex segmentoyotton, as birectified 8-simplex atop trirectified 8-simplex.

Vertex coordinates

The vertices of a trirectified 9-simplex of edge length 1 can be given in ten dimensions as all permutations of:

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

External links

• Klitzing, Richard. "treday".
• Wikipedia contributors. "Trirectified 9-simplex".