(Redirected from Jak)
Rank6
TypeUniform
SpaceSpherical
Notation
Bowers style acronymJak
Coxeter diagramx3o3o3o3o *c3o ()
Elements
Tera216+432 pentachora
Cells1080 tetrahedra
Faces720 triangles
Edges216
Vertices27
Vertex figureDemipenteract, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt6}{3} ≈ 0.81650}$
Hypervolume${\displaystyle \frac{\sqrt3}{16} ≈ 0.10825}$
Dipetal anglesTac–pen–hix: ${\displaystyle \arccos\left(-\frac{\sqrt6}{4}\right) ≈ 127.76124°}$
Tac–pen–tac: ${\displaystyle \arccos\left(-\frac14\right) ≈ 104.47751°}$
Central density1
Number of pieces99
Level of complexity3
Related polytopes
ArmyJak
RegimentJak
ConjugateNone
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryE6, order 51840
ConvexYes
NatureTame
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:

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

## 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.

o3o3o3o3o *c3o truncations
Name OBSA CD diagram Picture
Rectified pentacontatetrapeton ram
Pentacontatetrapeton mo
Truncated pentacontatetrapeton tim
Birectified pentacontatetrapeton barm
Small rhombated pentacontatetrapeton sram
Bitruncated pentacontatetrapeton bitem
Great rhombated pentacontatetrapeton gram