Icosiheptaheptacontadipeton

Icosiheptaheptacontadipeton
(OFF file)
Rank	6
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Jak
Coxeter diagram	x3o3o3o3o *c3o ()
Elements
Peta	72 hexatera, 27 triacontaditera
Tera	216+432 pentachora
Cells	1080 tetrahedra
Faces	720 triangles
Edges	216
Vertices	27
Vertex figure	Demipenteract, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt6}{3} ≈ 0.81650$
Hypervolume	$\frac{\sqrt3}{16} ≈ 0.10825$
Dipetal angles	Tac–pen–hix: $\arccos\left(-\frac{\sqrt6}{4}\right) ≈ 127.76124°$
	Tac–pen–tac: $\arccos\left(-\frac14\right) ≈ 104.47751°$
Central density	1
Number of pieces	99
Level of complexity	3
Related polytopes
Army	Jak
Regiment	Jak
Dual	Semistellatotriacontaditeric icosiheptapeton
Conjugate	None
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	E6, order 51840
Convex	Yes
Nature	Tame
Discovered by	{{{discoverer}}}

The icosiheptaheptacontadipeton, or jak, also called the 2₂₁ polytope, is a convex uniform polypeton. It has 27 triacontaditera and 72 hexatera as facets, with 10 triacontaditera and 16 hexatera at a vertex forming a demipenteract as the vertex figure.

The icosiheptaheptacontadipeton contains the vertices of a hexateric prism, and is also the convex hull of 3 gyro-orthogonal triangular duoprisms.

It can tile 6-dimensional Euclidean space by itself, forming the icosiheptaheptacontadipetic hexacomb.

Vertex coordinates

The vertices of an icosiheptaheptacontadipeton of edge length 1, centered at the origin, are given by:

• $\left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt6}{3}\right),$
• $\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt6}{12}\right)$ and all even sign changes of the first five coordinates,
• $\left(±\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,-\frac{\sqrt6}{6}\right)$ and all permutations of first 5 coordinates.

Representations

An icosiheptaheptacontadipeton has the following Coxeter diagrams:

• x3o3o3o3o *c3o (full symmetry)
• oox3ooo3ooo3oxo *c3ooo&#xt (D₅ axial, vertex-first)
• xox3ooo3ooo3oxo3ooo&#xt (A₅ axial, hexateron-first)
• xoxo ooox3oxoo3oooo3ooxo&#xt (A₄×A₁ symmetry, edge-first)
• xo3oo3oo3ox3oo xo&#zx (A₅×A₁ axial)
• xoo3oxo oxo3oox oox3xoo&#zx (A₂×A₂×A₂ symmetry, hull of 3 orthogonal triangular duoprisms)

Gallery

• 

  B₆ orthographic projection

• 

  D₅

• 

  D₄, A₂

• 

  A₅

• 

  A₄

• 

  A₃, D₃

Related polytopes

The icosiheptaheptacontadipeton is the colonel of a regiment with 9 uniform members, 5 fissary members, and one compound. Of its uniform members, three are noble.

External links

• Bowers, Jonathan. "Category 1: Primary Polypeta" (#154).

• Klitzing, Richard. "jak".

• Wikipedia Contributors. "2_21 polytope".