8-simplex

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8-simplex
Rank8
TypeRegular
Notation
Bowers style acronymEne
Coxeter diagramx3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3}
Tapertopic notation17
Elements
Zetta9 7-simplices
Exa36 6-simplices
Peta84 5-simplices
Tera126 pentachora
Cells126 tetrahedra
Faces84 triangles
Edges36
Vertices9
Vertex figure7-simplex, edge length 1
Measures (edge length 1)
Circumradius
Inradius
Hypervolume
Dizettal angle
Height
Central density1
Number of external pieces9
Level of complexity1
Related polytopes
ArmyEne
RegimentEne
DualEnneazetton
ConjugateNone
Abstract & topological properties
Flag count362880
Euler characteristic0
OrientableYes
Properties
SymmetryA8, order 362880
Flag orbits1
ConvexYes
NatureTame

The 8-simplex (also called the enneazetton, or ene) is the simplest possible non-degenerate 8-polytope. The full symmetry version has 9 regular 7-simplices as facets, joining 3 to a 6-simplex peak and 8 to a vertex, and is regular. It is the 8-dimensional simplex.

Vertex coordinates[edit | edit source]

The vertices of a regular 8-simplex of edge length 1, centered at the origin, are given by:

  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • .

Much simpler coordinates can be given in nine dimensions, as all permutations of:

  • .

Representations[edit | edit source]

A regular 8-simplex has the following Coxeter diagrams:

  • x3o3o3o3o3o3o3o () (full symmetry)
  • ox3oo3oo3oo3oo3oo3oo&#x (A7 axial, octaexal pyramid)

External links[edit | edit source]