# 8-simplex

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.

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${\displaystyle {\frac {2}{3}}\approx 0.66667}$
Inradius${\displaystyle {\frac {1}{12}}\approx 0.083333}$
Hypervolume${\displaystyle {\frac {1}{215040}}\approx 0.0000046503}$
Dizettal angle${\displaystyle \arccos \left({\frac {1}{8}}\right)\approx 82.81924^{\circ }}$
Height${\displaystyle {\frac {3}{4}}=0.75}$
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

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}}\right)}$ ,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}}\right)}$ ,
• ${\displaystyle \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}}\right)}$ ,
• ${\displaystyle \left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}}\right)}$ ,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}}\right)}$ ,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}}\right)}$ ,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}}\right)}$ ,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}}\right)}$ .

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

• ${\displaystyle \left({\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right)}$ .

## Representations

A regular 8-simplex has the following Coxeter diagrams:

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