Pentiruncicantic 6-cube

Pentiruncicantic 6-cube
Rank	6
Type	Uniform
Notation
Bowers style acronym	Cagrohax
Coxeter diagram	x3x3x3o3x *b3o ()
Elements
Peta	12 runcicantic 5-cubes, 32 runcicantellated 5-simplices, 32 stericantitruncated 5-simplices, 160 triangular-truncated tetrahedral duoprisms, 60 tesseractihexadecachoric prisms
Tera	640 triangular duoprisms, 640 triangular-hexagonal duoprisms, 960 truncated tetrahedral prisms, 240 cuboctahedral prisms, 480 truncated octahedral prisms, 192 decachora, 192 great rhombated pentachora, 192 prismatorhombated pentachora, 120 tesseractihexadecachora
Edges	20160
Vertices	5760
Vertex figure	Irregular square tettene
Measures (edge length 1)
Circumradius	${\frac {\sqrt {55}}{2}}\approx 3.70810$
Hypervolume	${\frac {762643}{120}}\approx 6355.35833$
Central density	1
Number of external pieces	296
Related polytopes
Army	Cagrohax
Regiment	Cagrohax
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	D6, order 23040
Convex	Yes
Nature	Tame

The pentiruncicantic 6-cube, also called the stericantitruncated 6-demicube, celligreatorhombidemihexeract or cagrohax, is a convex uniform 6-polytope. It consists of 12 runcicantic 5-cubes, 32 runcicantellated 5-simplices, 32 stericantitruncated 5-simplices, 160 triangular-truncated tetrahedral duoprisms, and 60 tesseractihexadecachoric prisms. 1 runcicantic 5-cube, 1 runcicantellated 5-simplex, 2 stericantitruncated 5-simplices, 1 triangular-truncated tetrahedral duoprism, and 2 tesseractihexadecachoric prisms join at each vertex. As the name suggests, it is the stericantitruncation of the 6-demicube.

Vertex coordinates

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

• $\left({\frac {7{\sqrt {8}}}{8}},\,{\frac {5{\sqrt {8}}}{8}},\,{\frac {5{\sqrt {8}}}{8}},\,{\frac {3{\sqrt {8}}}{8}},\,{\frac {\sqrt {8}}{8}},\,{\frac {\sqrt {8}}{8}}\right)$.

Representations

A pentiruncicantic 6-cube has the following Coxeter diagrams:

• x3x3x3o3x *b3o () (full symmetry)
• s4o3x3x3o3x () (as alternated pentiruncicantellated 6-cube)

Gallery

• 

  B₆ orthographic projection

• 

  D₅

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  D₄

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  D₃

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  A₅

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  A₃

External links

• Wikipedia contributors. "Pentiruncicantic 6-cube".