# Enneazetton

Enneazetton
Rank8
TypeRegular
SpaceSpherical
Notation
Bowers style acronymEne
Coxeter diagramx3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3}
Tapertopic notation17
Elements
Zetta9 octaexa
Exa36 heptapeta
Peta84 hexatera
Tera126 pentachora
Cells126 tetrahedra
Faces84 triangles
Edges36
Vertices9
Vertex figureOctaexon, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac23 ≈ 0.66667}$
Inradius${\displaystyle \frac{1}{12} ≈ 0.083333}$
Hypervolume${\displaystyle \frac{1}{215040} ≈ 0.0000046503}$
Dizettal angle${\displaystyle \arccos\left(\frac18\right) ≈ 82.81924^\circ}$
Height${\displaystyle \frac34 = 0.75}$
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
ConvexYes
NatureTame

The enneazetton, or ene, also commonly called the 8-simplex, is the simplest possible non-degenerate polyzetton. The full symmetry version has 9 regular octaexa as facets, joining 3 to a hexateron peak and 8 to a vertex, and is one of the 3 regular polyzetta. It is the 8-dimensional simplex.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),}$
• ${\displaystyle \left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),}$
• ${\displaystyle \left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23\right),}$

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

• ${\displaystyle \left(\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right).}$

## Representations

A regular enneazetton has the following Coxeter diagrams:

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