Birectified 1 22 polytope

Birectified 1 22 polytope
Rank	6
Type	Uniform
Notation
Bowers style acronym	Barm
Coxeter diagram	o3x3o3x3o *c3o ()
Elements
Peta	72 small birhombidodecatera 54 small rhombidemipenteracts
Tera	720 triangular duoprisms 432 rectified pentachora 864 small rhombated pentachora 270 rectified tesseracts
Cells	2160 tetrahedra 4320 triangular prisms 2160 octahedra 2160 cuboctahedra
Faces	4320+8640 triangles, 6480 squares
Edges	12960
Vertices	2160
Vertex figure	Irregular triangular-tetrahedral duoprism
Measures (edge length 1)
Circumradius	${\sqrt {6}}\approx 2.44949$
Hypervolume	${\frac {24267{\sqrt {3}}}{80}}\approx 525.39596$
Dipetal angles	Sirhin–srip–sibrid: $\arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }$
	Sibrid–triddip–sibrid: 120°
	Sirhin–rit–sirhin: 120°
	Sirhin–rap–sirhin: $\arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }$
Central density	1
Number of external pieces	126
Level of complexity	24
Related polytopes
Army	Barm
Regiment	Barm
Conjugate	None
Abstract & topological properties
Flag count	3110400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	E6×2, order 103680
Convex	Yes
Nature	Tame

The birectified 1 22 polytope or barm, also called the birectified 1₂₂ polytope, is a convex uniform polypeton. It consists of 54 small rhombidemipenteracts and 72 small birhombidodecatera. 4 small rhombidemipenteracts and 3 small birhombidodecatera join at each vertex. As the name suggests, it is the birectification of the pentacontatetrapeton.

Vertex coordinates

The vertices of a birectified 1 22 polytope of edge length 1, centered at the origin, are given by all permutations of the first five coordinates of

• $\left(\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,\pm {\frac {\sqrt {6}}{2}}\right)$,
• $\left(\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,0,\,0,\,0\right)$,
• $\left(\pm {\frac {3{\sqrt {2}}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right)$,

all permutations of the first five coordinates and all even sign changes of

• $\left({\frac {3{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {3{\sqrt {6}}}{4}}\right)$,

and all permutations of the first five coordinates and all odd sign changes of

• $\left({\frac {5{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {6}}{4}}\right)$.

Gallery

• 

  B₆ orthographic projection

• 

  D₅

• 

  D₄, A₂

• 

  A₅

• 

  A₄

• 

  A₃, D₃

Related polytopes

The birectified 1 22 polytope is the colonel of a regiment that includes 144 uniform members plus 13 fissaries.

External links

• Klitzing, Richard. "barm".
• Wikipedia contributors. "Birectified 1_22 polytope".