Birectified 3 21 polytope

Birectified 3 21 polytope
(OFF file)
Rank	7
Type	Uniform
Notation
Bowers style acronym	Branq
Coxeter diagram	o3o3o3o *c3x3o3o ()
Elements
Exa	576 birectified heptapeta, 126 birectified hexacontatetrapeta, 56 rectified icosiheptaheptacontadipeta
Peta	4032 rectified hexatera, 756 demipenteracts, 2016+4032 dodecatera, 1512 rectified triacontaditera
Tera	12096 pentachora, 7560 hexadecachora, 12096+12096+24192 rectified pentachora
Cells	10080+30240+60480 tetrahedra, 60480 octahedra
Faces	40320+120960 triangles
Edges	60480
Vertices	4032
Vertex figure	Triangular-rectified pentachoric duoprism, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {15}}{2}}\approx 1.93649$
Hypervolume	${\frac {56711}{280}}\approx 202.53929$
Diexal angles	Brag–dot–bril: $\arccos \left(-{\frac {2{\sqrt {7}}}{7}}\right)\approx 139.10661^{\circ }$
	Rojak–rix–bril: $\arccos \left(-{\frac {\sqrt {21}}{7}}\right)\approx 130.89339^{\circ }$
	Rojak–rat–brag: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Brag–dot–brag: 120°
	Rojak–hin–rojak: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Central density	1
Number of external pieces	758
Level of complexity	45
Related polytopes
Army	Branq
Regiment	Branq
Conjugate	None
Abstract & topological properties
Flag count	130636800
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	E7, order 2903040
Convex	Yes
Nature	Tame

The birectified hecatonicosihexapentacosiheptacontahexaexon, or branq, also called the birectified 3₂₁ polytope, is a convex uniform polyexon. It has 56 rectified icosiheptaheptacontadipeta, 126 birectified hexacontatetrapeta, and 576 birectified heptapeta. 3 rectified icosiheptaheptacontadipeta, 5 birectified hexacontatetrapeta, and 5 birectified heptapeta join at each triangular-rectified pentachoric duoprismatic vertex. As the name suggests, it is the birectification of the hecatonicosihexapentacosiheptacontahexaexon.

A birectified hecatonicosihexapentacosiheptacontahexaexon contains the vertices and edges of a small prismated hexacontatetrapetal prism, a terirhombated octaexon, a triangular-prismatotruncated hexateric duoprism, a hexagonal-cellirhombidodecateric duoprism, a cubic-rectified icositetrachoric duoprism, and a cuboctahedral-truncated octahedral duoprismatic prism.

Vertex coordinates

The vertices of a birectified hecatonicosihexapentacosiheptacontahexaexon of edge length 1, centered at the origin, are given by all permutations of first 6 coordinates of

• $\left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,\pm {\frac {1}{2}}\right),$

all permutations and even sign changes of the first 6 coordinates of

• $\left({\frac {3{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm 1\right),$
• $\left({\frac {5{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,0\right),$

and all permutations and odd sign changes of the first 6 coordinates of

• $\left({\frac {3{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,0\right).$

Representations

A birectified hecatonicosihexapentacosiheptacontahexaexon has the following Coxeter diagrams:

• o3o3o3o *c3x3o3o (full symmetry)
• oooxoo3xoxooo3oxooxo3oooxox3ooxooo *c3oooooo&#xt (E6 axial, rectified icosiheptaheptacontadipeton-first)
• oxo(xo)oxo3oox(oo)xoo3ooo(ox)ooo *b3xoo(ox)oox3oxo(oo)oxo3oox(uo)xoo&#xt (D6 axial, birectified hexacontatetrapeton-first)

External links

• Klitzing, Richard. "branq".
• Wikipedia contributors. "Birectified 3_21 polytope".