Icosiheptaheptacontadipeton

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)

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.