Pentiruncic 6-cube

Pentiruncic 6-cube
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Crohax
Coxeter diagram	x3o3x3o3x *b3o ()
Elements
Peta	160 triangular-tetrahedral duoprisms, 60 rectified tesseractic prisms, 32 small rhombated hexatera, 32 cellirhombidodecatera, 10 small rhombidemipenteracts
Tera	480+480 tetrahedral prisms, 640 triangular duoprisms, 192 rectified pentachora, 240+480 cuboctahedral prisms, 192+192 small rhombated pentachora, 120 rectified tesseracts
Cells	480+960 tetrahedra, 960+960+1920+1920 triangular prisms, 960 octahedra, 1440 cubes, 480+960 cuboctahedra
Faces	640+1920+1920+3840 triangles, 2880+2880+2880 squares
Edges	1920+2880+5760
Vertices	1920
Vertex figure	Irregular pentachoron atop triangular prism, edge lengths 1 (base triangular prism, triangle and top edge) and √2 (remaining edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {23}}{2}}\approx 2.39792$
Hypervolume	${\frac {159409}{360}}\approx 442.80278$
Dipetal angles	Rittip–tepe–tratet: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Sirhin–rit–rittip: 135°
	Card–triddip–tratet: 135°
	Sarx–srip–card: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
	Card–cope–rittip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Sarx–tepe–rittip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Sirhin–srip–card: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Sirhin–rap–sarx: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
Central density	1
Number of external pieces	296
Level of complexity	130
Related polytopes
Army	Crohax
Regiment	Crohax
Conjugate	None
Abstract & topological properties
Flag count	2995200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	D6, order 23040
Convex	Yes
Nature	Tame

The pentiruncic 6-cube, also called the stericantellated 6-demicube, cellirhombidemihexeract, or crohax, is a convex uniform 6-polytope. It consists of 12 runcic 5-cubes, 32 cantellated 5-simplices, 32 stericantellated 5-simplices, 160 triangular-tetrahedral duoprisms, and 60 rectified tesseractic prisms. 1 runcic 5-cube, 1 cantellated 5-simplex, 3 stericantellated 5-simplices, 1 triangular-tetrahedral duoprism, and 2 rectified tesseractic prisms join at each vertex. As the name suggests, it is the stericantellation of the 6-demicube.

Vertex coordinates

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

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

Representations

A pentiruncic 6-cube has the following Coxeter diagrams:

• x3o3x3o3x *b3o () (full symmetry)
• s4o3o3x3o3x () (as alternated penticantellated 6-orthoplex)

Gallery

• 

  B₆ orthographic projection

• 

  D₅

• 

  D₄

• 

  D₃

• 

  A₅

• 

  A₃

Related polytopes

THe regiment of the cellirhombidemihexeract contains 139 uniform members plus one fissary.

External links

• Klitzing, Richard. "crohax".
• Wikipedia contributors. "Pentiruncic 6-cube".