Runcinated 2 21 polytope

Runcinated 2 21 polytope
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Spojak
Coxeter diagram	x3o3o3x3o *c3x ()
Elements
Peta	720 triangular duoprismatic prisms, 216 small rhombated pentachoric prisms, 72 runcinated 5-simplices, 27 runcic 5-cubes, 27 runcinated 5-orthoplexes
Tera	1440 triangular duoprisms, 432 rectified pentachora, 2160+2160 triangular-square duoprisms, 1080 octahedral prisms, 216+432 small prismatodecachora, 1080 cuboctahedral prisms, 432 small rhombated pentachora, 270 rectified tesseracts
Cells	1080+2160 tetrahedra, 3×2160+3×4320 triangular prisms, 1080+2160 octahedra, 6480 cubes, 2160 cuboctahedra
Faces	3×4320+8640 triangles, 6480+6480+12960 squares
Edges	6480+6480+12960
Vertices	4320
Vertex figure	Triangle atop triangular fastegium, edge lengths 1 (triangle prism cell and opposing triangles) and √2 (all other edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {69}}{3}}\approx 2.76887$
Hypervolume	${\frac {119131{\sqrt {3}}}{160}}\approx 1289.63090$
Dipetal angles	Srippip–tisdip–tratrip: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$
	Spix–triddip–tratrip: 150°
	Sirhin–srip–srippip: $\arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }$
	Spix–ope–srippip: $\arccos \left(-{\frac {\sqrt {15}}{5}}\right)\approx 140.76848^{\circ }$
	Spat–tisdip–tratrip: 135°
	Spat–cope–srippip: $\arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }$
	Spat–spid–spix: $\arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }$
	Sirhin–rap–spix: $\arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }$
Central density	1
Number of external pieces	1062
Level of complexity	168
Related polytopes
Army	Spojak
Regiment	Spojak
Conjugate	None
Abstract & topological properties
Flag count	8709120
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	E6, order 51840
Convex	Yes
Nature	Tame

The runcinated 2₂₁ polytope, also called the small prismated icosiheptaheptacontadipeton or spojak, is a convex uniform 6-polytope. It consists of 27 runcinated 5-orthoplexes, 27 runcic 5-cubes, 72 runcinated 5-simplices, 216 small rhombated pentachoric prisms, and 720 triangular duoprismatic prisms. 2 runcinated 5-orthoplexes, 1 runcic 5-cube, 1 runcinated 5-simplex, 3 small rhombated pentachoric prisms, and 3 triangular duoprismatic prisms join at each vertex. As the name suggests, it is the runcination of the 2₂₁ polytope.

External links

• Klitzing, Richard. "spojak".