# Birectified 1 22 polytope

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Birectified 1 22 polytope
Rank6
TypeUniform
Notation
Bowers style acronymBarm
Coxeter diagramo3x3o3x3o *c3o ()
Elements
Peta72 small birhombidodecatera
54 small rhombidemipenteracts
Tera720 triangular duoprisms
432 rectified pentachora
864 small rhombated pentachora
270 rectified tesseracts
Cells2160 tetrahedra
4320 triangular prisms
2160 octahedra
2160 cuboctahedra
Faces4320+8640 triangles, 6480 squares
Edges12960
Vertices2160
Vertex figureIrregular triangular-tetrahedral duoprism
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {6}}\approx 2.44949}$
Hypervolume${\displaystyle {\frac {24267{\sqrt {3}}}{80}}\approx 525.39596}$
Dipetal anglesSirhin–srip–sibrid: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }}$
Sibrid–triddip–sibrid: 120°
Sirhin–rit–sirhin: 120°
Sirhin–rap–sirhin: ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$
Central density1
Number of external pieces126
Level of complexity24
Related polytopes
ArmyBarm
RegimentBarm
ConjugateNone
Abstract & topological properties
Flag count3110400
Euler characteristic0
OrientableYes
Properties
SymmetryE6×2, order 103680
ConvexYes
NatureTame

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

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

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

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

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

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

## Related polytopes

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