2 31 polytope

2 31 polytope
(OFF file)
Rank	7
Type	Uniform
Notation
Bowers style acronym	Laq
Coxeter diagram	x3o3o3o *c3o3o3o ()
Elements
Exa	576 heptapeta, 56 icosiheptaheptacontadipeta
Peta	4032 hexatera, 756 triacontaditera
Tera	4032+12096 pentachora
Cells	20160 tetrahedra
Faces	10080 triangles
Edges	2016
Vertices	126
Vertex figure	Demihexeract, edge length 1
Measures (edge length 1)
Circumradius	1
Hypervolume	${\frac {37}{70}}\approx 0.52857$
Diexal angles	Jak–hix–hop: $\arccos \left(-{\frac {\sqrt {21}}{7}}\right)\approx 130.89339^{\circ }$
	Jak–tac–jak: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Central density	1
Number of external pieces	632
Level of complexity	4
Related polytopes
Army	Laq
Regiment	Laq
Conjugate	None
Abstract & topological properties
Flag count	11612160
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	E7, order 2903040
Convex	Yes
Nature	Tame

The pentacontahexapentacosiheptacontahexaexon, or laq, also called the 2₃₁ polytope, is a convex uniform polyexon. It has 56 icosiheptaheptacontadipeta and 576 heptapeta as facets, with 12 icosiheptaheptacontadipeta and 32 heptapeta at a vertex forming a demihexeract as the vertex figure.

The pentacontahexapentacosiheptacontahexaexon contains the vertices of a demihexeractic prism, a tetrahedral duoprismatic prism, and a triangular-rectified hexateral duoprism, along with a hexadecaexon and small petihexadecaexon.

Vertex coordinates

The vertices of a pentacontahexapentacosiheptacontahexaexon of edge length 1, centered at the origin, are given by:

• $\left(0,\,0,\,0,\,0,\,0,\,0,\,\pm 1\right),$
• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm {\frac {1}{2}}\right)$ and all even sign changes of the first 6 coordinates
• $\left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0\right)$ and all permutations of first 6 coordinates

Representations

A pentacontahexapentacosiheptacontahexaexon has the following Coxeter diagrams:

• x3o3o3o *c3o3o3o (full symmetry)
• oxoxo3ooooo3ooooo *b3ooooo3ooxoo3ooooo&#xt (D6 axial, vertex-first)
• oxo3ooo3ooo *b3ooo3oox3ooo uxo&#zxt (D6×A1 symmetry)
• xoxoo3ooooo3oooxo3oxooo3ooooo3ooxox&#xt (A6 axial, heptapeton-first)
• xoo3ooo3ooo3ooo3oox *c3oxo&#xt (E6 axial, icosiheptaheptacontadipeton-first)
• xo3oo3oo3ox3oo3oo3xo&#zx (A7 symmetry)
• ox(uo)xo ox(oo)oo3oo(oo)oo3oo(oo)xo *c3oo(ox)oo3xo(oo)ox&#xt (D5×A1 axial)

External links

• Klitzing, Richard. "laq".
• Wikipedia contributors. "2_31 polytope".