Enneazetton

Enneazetton
(OFF file)
Rank	8
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Ene
Coxeter diagram	x3o3o3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3,3,3}
Tapertopic notation	17
Elements
Zetta	9 octaexa
Exa	36 heptapeta
Peta	84 hexatera
Tera	126 pentachora
Cells	126 tetrahedra
Faces	84 triangles
Edges	36
Vertices	9
Vertex figure	Octaexon, edge length 1
Measures (edge length 1)
Circumradius	$\frac23 ≈ 0.66667$
Inradius	$\frac{1}{12} ≈ 0.083333$
Hypervolume	$\frac{1}{215040} ≈ 0.0000046503$
Dizettal angle	$\arccos\left(\frac18\right) ≈ 82.81924^\circ$
Height	$\frac34 = 0.75$
Central density	1
Number of external pieces	9
Level of complexity	1
Related polytopes
Army	Ene
Regiment	Ene
Dual	Enneazetton
Conjugate	None
Abstract & topological properties
Flag count	362880
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A8, order 362880
Convex	Yes
Nature	Tame

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:

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

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

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

Representations

A regular enneazetton has the following Coxeter diagrams:

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

External links

• Klitzing, Richard. "ene".

• Wikipedia Contributors. "8-simplex".