Bitruncated 6-cube

Bitruncated 6-cube
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Botox
Coxeter diagram	o4x3x3o3o3o ()
Elements
Peta	64 truncated 5-simplices, 12 bitruncated 5-cubes
Tera	192 pentachora, 384 truncated pentachora, 60 tesseractihexadecachora
Cells	960 tetrahedra, 960 truncated tetrahedra, 160 truncated octahedra
Faces	1920 triangles, 240 squares, 1280 hexagons
Edges	960+1920
Vertices	960
Vertex figure	Tetrahedral scalene, edge lengths 1 (base tetrahedron), √2 (top edge), and √3 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {34}}{2}}\approx 2.91548$
Hypervolume	${\frac {15119}{30}}\approx 503.96667$
Dipetal angles	Tix–pen–tix: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
	Bittin–tip–tix: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Bittin–tah–bittin: 90°
Central density	1
Number of external pieces	76
Level of complexity	15
Related polytopes
Army	Botox
Regiment	Botox
Conjugate	None
Abstract & topological properties
Flag count	691200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B6, order 46080
Convex	Yes
Nature	Tame

The bitruncated 6-cube, also called the bitruncated hexeract or botox, is a convex uniform 6-polytope. It consists of 12 bitruncated 5-cubes and 64 truncated 5-simplices. 4 bitruncated 5-cubes and 2 truncated 5-simplices join at each vertex. As the name suggests, it is the bitruncation of the 6-cube.

Vertex coordinates

The vertices of a bitruncated 6-cube of edge length 1 are given by all permutations of:

• $\left(\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right)$.

Representations

A bitruncated 6-cube has the following Coxeter diagrams:

• o4x3x3o3o3o () (full symmetry)
• x3x3o3o3o *b3x () (D₆ symmetry)

Gallery

• 

  B₅ orthographic projection

• 

  B₄

• 

  B₃

• 

  B₂

• 

  A₅

• 

  A₃

External links

• Klitzing, Richard. "botox".
• Wikipedia contributors. "Bitruncated 6-cube".