# Octaexon

Octaexon Rank7
TypeRegular
SpaceSpherical
Notation
Bowers style acronymOca
Coxeter diagramx3o3o3o3o3o3o (             )
Schläfli symbol{3,3,3,3,3,3}
Tapertopic notation16
Elements
Exa8 heptapeta
Peta28 hexatera
Tera56 pentachora
Cells70 tetrahedra
Faces56 triangles
Edges28
Vertices8
Vertex figureHeptapeton, edge length 1
Measures (edge length 1)
Circumradius$\frac{\sqrt7}{4} ≈ 0.66144$ Edge radius$\frac{\sqrt3}{4} \approx 0.43301$ Face radius$\frac{\sqrt{15}}{12} ≈ 0.32275$ Cell radius$\frac14 = 0.25$ Teron radius$\frac{\sqrt{15}}{20} ≈ 0.19365$ Peton radius$\frac{\sqrt3}{12} ≈ 0.14434$ Inradius$\frac{\sqrt7}{28} ≈ 0.094491$ Hypervolume$\frac{1}{20160} ≈ 0.000049603$ Diexal angle$\arccos\left(\frac17\right) ≈ 81.78679^\circ$ HeightsPoint atop hop: $\frac{2\sqrt7}{7} ≈ 0.75593$ Dyad atop perp hix: $\frac{\sqrt3}{3} ≈ 0.57735$ Trig atop perp pen: $\frac{2\sqrt{15}}{15} ≈ 0.51640$ Tet atop perp tet: $\frac12 = 0.5$ Central density1
Number of external pieces8
Level of complexity1
Related polytopes
ArmyOca
RegimentOca
DualOctaexon
ConjugateNone
Abstract & topological properties
Flag count40320
Euler characteristic2
OrientableYes
Properties
SymmetryA7, order 40320
ConvexYes
NatureTame

The octaexon, or oca, also commonly called the 7-simplex, is the simplest possible non-degenerate polyexon. The full symmetry version has 8 regular heptapeta as facets, joining 3 to a pentachoron peak and 7 to a vertex, and is one of the 3 regular polyexa. It is the 7-dimensional simplex. It is also a pyramid based on the heptapeton.

A regular octaexon of edge length 2 can be inscribed in the unit hepteract. The next largest simplex that can be inscribed in a hypercube is the dodecadakon.

## Vertex coordinates

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

• $\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$ • $\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$ • $\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$ • $\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$ • $\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),$ • $\left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28}\right),$ • $\left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4}\right).$ Much simpler sets of coordinates can be found by inscribing the octaexon into the hepteract. One such set is given by:

• $\left(\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14\right),$ • $\left(\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,-\frac14\right),$ • $\left(\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,\frac14\right),$ • $\left(\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14\right),$ • $\left(-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,-\frac14\right),$ • $\left(-\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14\right),$ • $\left(-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,\frac14\right),$ • $\left(-\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14\right).$ Even simpler coordinates can be given in eight dimensions, as all permutations of:

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

An octaexon has the following Coxeter diagrams:

• x3o3o3o3o3o3o (full symmetry)
• ox3oo3oo3oo3oo3oo&#x (A6 axial, heptapetal pyramid)
• xo ox3oo3oo3oo3oo&#x (A5×A1 axial, hexateral scalene)
• xo3oo ox3oo3oo&#x (A4×A2 axial, pentachoric tettene)
• xo3oo3oo ox3oo3oo&#x (A3×A3 axial, tetrahedral disphenoid)
• oxo3ooo oox3ooo3ooo&#x (A3×A2 symmetry, tetrahedral tettene pyramid)
• oxo xoo3ooo ooxooo&#x (A2×A2×A1 symmetry, trigonal disphenoid scalene)
• xoo oox oxo3ooo3ooo&#x (A3×A1×A1 symmetry, tetrahedral scalenic scalene)
• oxoo3oooo ooxo3oooo&#x (A2×A2 symmetry, trigonal pyramidal disphenoid)