1 32 polytope

1 32 polytope
(OFF file)
Rank	7
Type	Uniform
Notation
Bowers style acronym	Lin
Coxeter diagram	o3o3o3x *c3o3o3o ()
Elements
Exa	126 demihexeracts, 56 pentacontatetrapeta
Peta	2016 hexatera, 756+1512 demipenteracts
Tera	4032+12096 pentachora, 7560 hexadecachora
Cells	20160+30240 tetrahedra
Faces	40320 triangles
Edges	10080
Vertices	576
Vertex figure	Birectified heptapeton, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {7}}{2}}\pm 1.32288$
Hypervolume	8
Diexal angles	Mo–hin–hax: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Hax–hix–hax: 120º
	Mo–hin–mo: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Central density	1
Number of external pieces	182
Level of complexity	10
Related polytopes
Army	Lin
Regiment	Lin
Conjugate	None
Abstract & topological properties
Flag count	29030400
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	E7, order 2903040
Convex	Yes
Nature	Tame

The pentacontahexahecatonicosihexaexon, or lin, also called the 1₃₂ polytope, is a convex uniform polyexon. It has 56 pentacontatetrapeta and 126 demihexeracts as facets, with 7 of each at a vertex forming a birectified heptapeton as the vertex figure.

The pentacontahexahecatonicosihexaexon contains the vertices of a birectified hexacontatetrapetal prism, a cuboctahedral duoprismatic prism, a cubic-icositetrachoric duoprism, a hepteract, a small cellated octaexon, a triangular-small prismated hexateric duoprism, and a hexagonal-dodecateric duoprism.

It can tile 7-dimensional Euclidean space by itself, forming the pentacontahexahecatonicosihexaexic heptacomb.

Vertex coordinates

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

• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm 1\right)$ and all even sign changes of the first 6 coordinates
• $\left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,\pm {\frac {1}{2}}\right)$ and all permutations of first 6 coordinates
• $\left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,0\right)$ and all odd sign changes and all permutations of the first 6 coordinates

Representations

A pentacontahexahecatonicosihexaexon has the following Coxeter diagrams:

• o3o3o3x *c3o3o3o (full symmetry)
• oooo3oxoo3oooo3ooxo3oooo *c3xoox&#xt (E6 axial, pentacontatetrapeton-first)
• xooox3ooooo3ooxoo *b3oxoxo3ooooo3ooxoo&#xt (D6 axial, demihexeract-first)

External links

• Klitzing, Richard. "lin".
• Wikipedia contributors. "1_32 polytope".