Bitruncated 3 21 polytope

Bitruncated 3 21 polytope
File:Up2 3 21 t12 E7.svg (OFF file)
Rank	7
Type	Uniform
Notation
Bowers style acronym	Botnaq
Coxeter diagram	o3o3o3o *c3x3x3o ()
Elements
Exa	576 bitruncated heptapeta, 56 truncated icosiheptaheptacontadipeta, 126 bitruncated hexacontatetrapeta
Peta	756 demipenteracts, 4032 truncated hexatera, 1512 truncated triacontaditera, 2016+4032 bitruncated hexatera
Tera	12096 pentachora, 7560 hexadecachora, 12096+24192 truncated pentachora, 12096 decachora
Cells	30240+60480 tetrahedra, 10080+60480 truncated tetrahedra
Faces	4032+120960 triangles, 40320 hexagons
Edges	12096+60480
Vertices	12096
Vertex figure	Rectified-pentachoric scalene, edge lengths 1 (base rectified pentachoron and top edge) and √3 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {43}}{2}}\approx 3.27873$
Hypervolume	${\frac {2350343}{120}}\approx 19586.19167$
Diexal angles	Botag–bittix–batal: $\arccos \left(-{\frac {2{\sqrt {7}}}{7}}\right)\approx 139.10661^{\circ }$
	Tojak–tix–batal: $\arccos \left(-{\frac {\sqrt {21}}{7}}\right)\approx 130.89339^{\circ }$
	Tojak–tot–botag: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Botag–bittix–botag: 120°
	Tojak–hin–tojak: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Central density	1
Number of external pieces	758
Level of complexity	63
Related polytopes
Army	Botnaq
Regiment	Botnaq
Conjugate	None
Abstract & topological properties
Flag count	182891520
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	E7, order 2903040
Convex	Yes
Nature	Tame

The bitruncated hecatonicosihexapentacosiheptacontahexaexon, or botnaq, also called the bitruncated 3₂₁ polytope, is a convex uniform polyexon. It has 56 truncated icosiheptaheptacontadipeta, 576 bitruncated heptapeta, and 126 bitruncated hexacontatetrapeta. 2 truncated icosiheptaheptacontadipeta, 5 bitruncated heptapeta, and 5 bitruncated hexacontatetrapeta join at each vertex. As the name suggests, it is the bitruncation of the hecatonicosihexapentacosiheptacontahexaexon.

Vertex coordinates

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

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

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

• $\left({\frac {5{\sqrt {2}}}{4}},\,{\frac {5{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm 2\right),$
• $\left({\frac {3{\sqrt {2}}}{2}},\,{\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,\pm {\frac {3}{2}}\right),$
• $\left({\frac {7{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm 1\right),$
• $\left(2{\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {9{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,0\right),$

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

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

External links

• Klitzing, Richard. "botnaq".