2 21 polytope

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221 polytope
Rank6
TypeUniform
Notation
Bowers style acronymJak
Coxeter diagramx3o3o3o3o *c3o ()
Elements
Peta72 5-simplices, 27 5-orthoplexes
Tera216+432 pentachora
Cells1080 tetrahedra
Faces720 triangles
Edges216
Vertices27
Vertex figureDemipenteract, edge length 1
Measures (edge length 1)
Circumradius
Hypervolume
Dipetal anglesTac–pen–hix:
 Tac–pen–tac:
Central density1
Number of external pieces99
Level of complexity3
Related polytopes
ArmyJak
RegimentJak
DualSemistellatotriacontaditeric icosiheptapeton
ConjugateNone
Abstract & topological properties
Flag count155520
Euler characteristic0
OrientableYes
Properties
SymmetryE6, order 51840
ConvexYes
NatureTame


The 221 polytope (also called the icosiheptaheptacontadipeton or jak) is a convex uniform 6-polytope. It has 27 5-orthoplexes and 72 5-simplices as facets, with 10 5-orthoplexes and 16 5-simplices at each vertex forming a demipenteract as the vertex figure.

The 221 polytope 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 222 honeycomb.

Vertex coordinates[edit | edit source]

The vertices of a 221 polytope of edge length 1, centered at the origin, are given by:

  • ,
  • and all even sign changes of the first five coordinates,
  • and all permutations of first 5 coordinates.

Representations[edit | edit source]

A 221 polytope 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[edit | edit source]

Related polytopes[edit | edit source]

The 221 polytope is the colonel of a regiment with 9 uniform members, 5 fissary members, and one compound. Of its uniform members, three are noble.

It is the real analog of the Hessian polyhedron.

External links[edit | edit source]