Steriruncic 6-cube

Steriruncic 6-cube
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Prohax
Coxeter diagram	x3o3x3x3o *b3o ()
Elements
Peta	160 triangular-tetrahedral duoprisms, 32 bitruncated hexatera, 32 prismatorhombated hexatera, 12 prismatorhombidemipenteracts
Tera	480 tetrahedral prisms, 640 triangular duoprisms, 192 truncated pentachora, 240 truncated tetrahedral prisms, 192 decachora, 60 rectified tesseracts, 192 prismatorhombated pentachora
Cells	480+480 tetrahedra, 960+1920 triangular prisms, 960 hexagonal prisms, 480+960 truncated tetrahedra, 480 cuboctahedra
Faces	640+1920+1920 triangles, 1440+2880 squares, 1920 hexagons
Edges	1920+2880+2880
Vertices	1920
Vertex figure	Skewed triangular-prismatic scalene, base edge lengths 1 and √2, top edge length 1, lacing lengths √2 and √3
Measures (edge length 1)
Circumradius	${\frac {\sqrt {31}}{2}}\approx 2.78388$
Hypervolume	${\frac {106513}{120}}\approx 887.60833$
Dipetal angles	Pirhin–tepe–tratet: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Pirx–triddip–tratet: 135°
	Bittix–deca–pirx: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
	Pirhin–prip–pirx: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Pirhin–tip–bittix: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Pirx–tuttip–pirx: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	Pirhin–rit–pirhin: 90°
Central density	1
Number of external pieces	236
Level of complexity	90
Related polytopes
Army	Prohax
Regiment	Prohax
Conjugate	None
Abstract & topological properties
Flag count	2073600
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	D6, order 23040
Convex	Yes
Nature	Tame

The steriruncic 6-cube, also called the runcicantellated 6-demicube, prismatorhombidemihexeract, or prohax, is a convex uniform 6-polytope. It consists of 12 steriruncic 5-cubes, 32 bitruncated 5-simplices, 32 runcicantellated 5-simplices, and 160 triangular-tetrahedral duoprisms. 2 steriruncic 5-cubes, 1 bitruncated 5-simplex, 3 runcicantellated 5-simplices, and 1 triangular-tetrahedral duoprism join at each vertex. As the name suggests, it is the runcicantellation of the 6-demicube.

Vertex coordinates

The coordinates of a steriruncic 6-cube, centered at the origin and with unit edge length, are given by all permutations and even sign changes of:

• $\left({\frac {5{\sqrt {2}}}{4}},\,{\frac {5{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)$.

Representations

A steriruncic 6-cube has the following Coxeter diagrams:

• x3o3x3x3o *b3o () (full symmetry)
• s4o3o3x3x3o () (as alternated steriruncinated 6-cube)

Gallery

• 

  B₆ orthographic projection

• 

  D₅

• 

  D₄

• 

  D₃

• 

  A₅

• 

  A₃

Related polytopes

The regiment of the prismatorhombidemihexeract contains 7 uniform members.

External links

• Klitzing, Richard. "prohax".
• Wikipedia contributors. "Steriruncic 6-cube".