4 21 polytope

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421 polytope
Rank8
TypeUniform
Notation
Bowers style acronymFy
Coxeter diagramo3o3o3o3o3o3x *c3o ()
Elements
Zetta17280 7-simplices, 2160 7-orthoplexes
Exa69120+138240 6-simplices
Peta483840 5-simplices
Tera483840 pentachora
Cells241920 tetrahedra
Faces60480 triangles
Edges6720
Vertices240
Vertex figure321 polytope, edge length 1
Measures (edge length 1)
Circumradius1
Hypervolume
Dizettal anglesoca–hop-zee:
 zee–hop-zee:
Central density1
Number of external pieces19440
Level of complexity3
Related polytopes
ArmyFy
RegimentFy
DualSemistellatotriacontaditeri-icosiheptapeto-pentacontoctaexic diacositetracontazetton
ConjugateNone
Abstract & topological properties
Flag count2090188800
Euler characteristic0
OrientableYes
Properties
SymmetryE8, order 696729600
ConvexYes
NatureTame


The 421 polytope (also called dischiliahecatonhexaconta-myriaheptachiliadiacosioctaconta-zetton, abbreviated as 2160-17280-zetton, fy, or E8 polytope) or the is a uniform 8-polytope. It consists of 2160 7-orthoplexes and 17280 7-simplices as facets. At each vertex 126 7-orthoplexes and 576 7-simplexes meet forming a 321 polytope as the vertex figure.

The 421 polytope contains the vertices and edges of the birectified 8-simplex, small exioctadecazetton, demiocteract, rectified 8-orthoplex, 321 polytope prism, triangular-221 polytope prism, pentachoric-rectified pentachoric duoprism, tetrahedral-demipenteractic duoprism, hexadecachoric duoprism, octahedral-triacontiditeric duoprism, square-hexacontatetrapetic duoprism, rectified octaexic prism, and triangular-hexateric duoprismatic prism.

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a 421 polytope with edge length 1 are given by all permutations and sign changes of

  • ,

as well as all even sign changes of

  • .

Integral coordinates can be given in 9 dimensions, as all permutations of:

  • ,
  • ,
  • .

These give an edge length , and come from the fact that it is the convex hull of an expanded 8-simplex and two opposite birectified 8-simplices.

Representations[edit | edit source]

A 421 polytope has the following Coxeter diagrams:

  • o3o3o3o3o3o3x *c3o () (full symmetry)
  • ooxoo3ooooo3ooooo3ooooo *c3ooooo3ooooo3oxoxo&#xt (E7 axial, vertex-first)
  • oxooo3ooooo3oooxo *b3ooooo3ooooo3ooxoo3xooox&#xt (D7 axial, hecatonicosoctaexon-first)
  • xo3oo3oo *b3oo3oo3oo3ox3oo&#zx (D8 subsymmetry)
  • oxo3ooo3xoo3ooo3ooo3oox3ooo3oxo&#zx (A8 subsymmetry)
  • xooxooo3oooooxo3ooxoooo3ooooooo3ooooxoo3oxooooo3oooxoox&#xt (A7 axial, octaexon-first)
  • xoxooo3ooooxo3ooooox3xooxoo oxooox3ooxooo3oooxoo3oxooxo&#zx (A4×A4 subsymmetry)
  • ooxoo3xoooo3oooxo *b3oooox ooxoo3oxooo3oooxo *f3oooox&#zx (D4×D4 subsymmetry)
  • xoxo3xoox ooxo3oooo3oooo3oooo3ooox *e3oxoo&#zx (A6×A2 subsymmetry)

Gallery[edit | edit source]

External links[edit | edit source]