Biruncitruncated 6-cube

Biruncitruncated 6-cube
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Boprag
Coxeter diagram	o4x3x3o3x3o ()
Elements
Peta	240 square-octahedral duoprisms 160 triangular-truncated octahedral duoprisms 64 runcitruncated 5-simplices 12 runcicantellated 5-orthoplexes
Tera	960+960 triangular-square duoprisms 960 octahedral prisms 1280 triangular-hexagonal duoprisms 192 small rhombated pentachora 480 truncated octahedral prisms 384 prismatorhombated pentachora 60 tesseractihexadecachora
Cells	1920+3840+3840 triangular prisms 960 octahedra 2880 cubes 3840 hexagonal prisms 960 truncated tetrahedra 960 cuboctahedra 480 truncated octahedra
Faces	1920+3840+3840 triangles 1440+5760+11520 squares 3840 hexagons
Edges	5760+5760+11520
Vertices	5760
Vertex figure	Wedge scalene, edge lengths 1 (base square and top edge of wedge), √2 (sides of wedge and lacings to square, and top edge between apices), and √3 (lacings to top edge)
Measures (edge length 1)
Circumradius	${\frac {3{\sqrt {6}}}{2}}\approx 3.67423$
Hypervolume	${\frac {422887}{90}}\approx 4698.74444$
Dipetal angles	Squoct–tisdip–tratoe: 150°
	Pattix–ope–squoct: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Pirt–tope–tratoe: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Pattix–thiddip–tratoe: 135°
	Pattix–srip–pattix: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
	Pirt–tisdip–squoct: 120°
	Pirt–prip–pattix: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Pirt–tah–pirt: 90°
Central density	1
Number of external pieces	476
Level of complexity	135
Related polytopes
Army	Boprag
Regiment	Boprag
Conjugate	None
Abstract & topological properties
Flag count	6220800
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B6, order 46080
Convex	Yes
Nature	Tame

The biruncitruncated 6-cube, also called the biruncitruncated hexacontatetrapeton, biprismatorhombated hexacontatetrapeton, or boprag, is a convex uniform 6-polytope. It consists of 12 runcicantellated 5-orthoplexes, 64 runcitruncated 5-simplices, 240 square-octahedral duoprisms, and 160 triangular-truncated octahedral duoprisms. 2 runcicantellated 5-orthoplexes, 2 runcitruncated 5-simplices, 1 square-octahedral duoprism, and 2 triangular-truncated octahedral duoprisms join at each vertex. As the name suggests, it is the biruncitruncation of the 6-orthoplex.

Vertex coordinates

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

• $\left(\pm {\frac {3{\sqrt {2}}}{2}},\,\pm {\frac {3{\sqrt {2}}}{2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right)$.

Representations

A biruncitruncated 6-cube has the following Coxeter diagrams:

• o4x3x3o3x3o () (full symmetry)
• x3x3o3x3o *b3x () (D₆ symmetry)

Gallery

• 

  B₅ orthographic projection

• 

  B₄

• 

  B₃

• 

  B₂

• 

  A₅

• 

  A₃

Related polytopes

The biprismatorhombated hexacontatetrapeton is the colonel of a regiment that includes 7 uniform members.

External links

• Klitzing, Richard. "boprag".
• Wikipedia contributors. "Biruncitruncated 6-cube".