3 21 polytope

3 21 polytope
(OFF file)
Rank	7
Type	Uniform
Notation
Bowers style acronym	Naq
Coxeter diagram	o3o3o3o *c3o3o3x ()
Elements
Exa	576 heptapeta, 126 hexacontatetrapeta
Peta	2016+4032 hexatera
Tera	12096 pentachora
Cells	10080 tetrahedra
Faces	4032 triangles
Edges	756
Vertices	56
Vertex figure	Icosiheptaheptacontadipeton, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Hypervolume	${\frac {17}{140}}\approx 0.12143$
Diexal angles	Gee–hix–hop: $\arccos \left(-{\frac {2{\sqrt {7}}}{7}}\right)\approx 139.10661^{\circ }$
	Gee–hix–gee: 120°
Central density	1
Number of external pieces	702
Level of complexity	3
Related polytopes
Army	Naq
Regiment	Naq
Dual	Pentacontahexaexon
Conjugate	None
Abstract & topological properties
Flag count	8709120
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	E7, order 2903040
Convex	Yes
Nature	Tame

The hecatonicosihexapentacosiheptacontahexaexon, or naq, also called the 3₂₁ polytope or biambohexadecaexon, is a convex uniform polyexon. It has 126 hexacontatetrapeta and 576 heptapeta as facets, with 27 hexacontatetrapeta and 72 heptapeta at a vertex forming an icosiheptaheptacontadipeton as the vertex figure.

The hecatonicosihexapentacosiheptacontahexaexon contains the vertices of a hexacontatetrapetic prism and a triangular-hexateric duoprism. It is also the convex hull of two oppositely oriented rectified octaexa.

Vertex coordinates

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

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

Alternate coordinates can be given in 8 dimensions, as all permutations of:

• $\pm \left(3,\,3,\,-1,\,-1,\,-1,\,-1,\,-1,\,-1\right)$.

These give a naq of edge length $4{\sqrt {2}}$, and come from the fact that naq is the convex hull of two opposite rectified octaexa.

Representations

A hecatonicosihexapentacosiheptacontahexaexon has the following Coxeter diagrams:

• o3o3o3o *c3o3o3x (full symmetry)
• oxo3ooo3ooo *b3ooo3ooo3xox&#xt (D6 axial, hexacontatetrapeton-first)
• oxoo3oooo3oooo3oooo3ooxo *c3oooo&#xt (E6 axial, vertex-first)
• xoxox oxooo3ooooo3oooxo *c3ooooo3ooxoo&#xt (D5×A1 axial, edge-first)
• xooo3ooxo3oooo3oooo3oxoo3ooox&#xt (A6 axial, heptapeton-first)
• oo3xo3oo3oo3oo3ox3oo&#zx (A7 symmetry, hull of 2 rectified octaexa)
• ox xo3oo3oo *c3oo3oo3ox&#zx (D6×A1 symmetry)
• xxoo xoxo xoox oxoo3oooo3ooxo *e3ooox&#zx (D4×A1×A1×A1 symmetry)

External links

• Klitzing, Richard. "naq".
• Wikipedia contributors. "3_21 polytope".