Stericantic 6-cube

Stericantic 6-cube
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Pithax
Coxeter diagram	x3x3o3x3o *b3o ()
Elements
Peta	160 triangular-truncated tetrahedral duoprisms, 32 small birhombidodecatera, 32 prismatotruncated hexatera, 12 prismatotruncatodemipenteracts
Tera	640 triangular duoprisms, 240 octahedral prisms, 640 triangular-hexagonal duoprisms, 480 truncated tetrahedral prisms, 192+192 small rhombated pentachora, 60 truncated hexadecachora, 192 prismatorhombated pentachora
Cells	960+960+1920+1920 triangular prisms, 480+480 octahedra, 1920 hexagonal prisms, 480+480 truncated tetrahedra, 960 cuboctahedra
Faces	1920+1920+1920+1920 triangles, 2880+5760 squares, 1920 hexagons
Edges	1440+5760+5760
Vertices	2880
Vertex figure	Skewed tetragonal disphenoidal prismatic pyramid, edge lengths 1 (base squares), √2 (lacings to one base and lacings of disphenoids), and √3 (lacings to other base)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {35}}{2}}\approx 2.95804$
Hypervolume	${\frac {468859}{360}}\approx 1302.38611$
Dipetal angles	Pithin–tuttip–tratut: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Sibrid–triddip–tratut: 135°
	Pattix–thiddip–tratut: 135°
	Sibrid–srip–pattix: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
	Pithin–prip–pattix: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Pithin–srip–sibrid: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Pattix–ope–pattix: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	Pithin–thex–pithin: 90°
Central density	1
Number of external pieces	236
Level of complexity	144
Related polytopes
Army	Pithax
Regiment	Pithax
Conjugate	None
Abstract & topological properties
Flag count	3317760
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	D6, order 23040
Convex	Yes
Nature	Tame

The stericantic 6-cube, also called the runcitruncated 6-demicube, prismatotruncatodemihexeract, or pithax, is a convex uniform 6-polytope. It consists of 12 stericantic 5-cubes, 32 bicantellated 5-simplices, 32 runcitruncated 5-simplices, and 160 triangular-truncated tetrahedral duoprisms. 2 stericantic 5-cubes, 1 bicantellated 5-simplex, 2 runcitruncated 5-simplices, and 2 triangular-truncated tetrahedral duoprisms join at each vertex. As the name suggests, it is the runcitruncation of the 6-demicube.

Vertex coordinates

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

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

Representations

A stericantic 6-cube has the following Coxeter diagrams:

• x3x3o3x3o *b3o () (full symmetry)
• s4o3x3o3x3o () (as alternated stericantellated 6-cube)

Gallery

• 

  B₆ orthographic projection

• 

  D₅

• 

  D₄

• 

  D₃

• 

  A₅

• 

  A₃

Related polytopes

The regiment of the prismatotruncatodemihexeract contains 31 uniform members.

External links

• Klitzing, Richard. "pithax".
• Wikipedia contributors. "Stericantic 6-cube".