# 7-simplex

7-simplex
Rank7
TypeRegular
Notation
Bowers style acronymOca
Coxeter diagramx3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3}
Tapertopic notation16
Elements
Exa8 6-simplices
Peta28 5-simplices
Tera56 pentachora
Cells70 tetrahedra
Faces56 triangles
Edges28
Vertices8
Vertex figure6-simplex, edge length 1
Measures (edge length 1)
Circumradius$\displaystyle \frac{\sqrt7}{4} \approx 0.66144$
Edge radius$\displaystyle \frac{\sqrt3}{4} \approx 0.43301$
Face radius$\displaystyle \frac{\sqrt{15}}{12} \approx 0.32275$
Cell radius$\displaystyle \frac14 = 0.25$
Teron radius$\displaystyle \frac{\sqrt{15}}{20} \approx 0.19365$
Peton radius$\displaystyle \frac{\sqrt3}{12} \approx 0.14434$
Inradius$\displaystyle \frac{\sqrt7}{28} \approx 0.094491$
Hypervolume$\displaystyle \frac{1}{20160} \approx 0.000049603$
Diexal angle$\displaystyle \arccos\left(\frac17\right) \approx 81.78679^\circ$
HeightsPoint atop hop: $\displaystyle \frac{2\sqrt7}{7} \approx 0.75593$
Dyad atop perp hix: $\displaystyle \frac{\sqrt3}{3} \approx 0.57735$
Trig atop perp pen: $\displaystyle \frac{2\sqrt{15}}{15} \approx 0.51640$
Tet atop perp tet: $\displaystyle \frac12 = 0.5$
Central density1
Number of external pieces8
Level of complexity1
Related polytopes
ArmyOca
RegimentOca
DualOctaexon
ConjugateNone
Abstract & topological properties
Flag count40320
Euler characteristic2
OrientableYes
SkeletonK 8
Properties
SymmetryA7, order 40320
Flag orbits1
ConvexYes
NatureTame

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

A regular 7-simplex of edge length 2 can be inscribed in the unit hepteract.[1] The next largest simplex that can be inscribed in a hypercube is the dodecadakon.[2]

## Vertex coordinates

The vertices of a regular 7-simplex of edge length 1, centered at the origin, are given by:

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

Much simpler sets of coordinates can be found by inscribing the 7-simplex into the hepteract. One such set is given by:[3]

• $\displaystyle \left(\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14\right)$ ,
• $\displaystyle \left(\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,-\frac14\right)$ ,
• $\displaystyle \left(\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,\frac14\right)$ ,
• $\displaystyle \left(\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14\right)$ ,
• $\displaystyle \left(-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,-\frac14\right)$ ,
• $\displaystyle \left(-\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14\right)$ ,
• $\displaystyle \left(-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,\frac14\right)$ ,
• $\displaystyle \left(-\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14\right)$ .

Even simpler coordinates can be given in eight dimensions, as all permutations of:

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

## Representations

A 7-simplex 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)

## References

1. Adams, Joshua; Zvengrowski, Peter; Laird, Philip (2003). "Vertex Embeddings of Regular Polytopes". Expositiones Mathematicae.
2. Sloane, N. J. A. (ed.). "Sequence A019442". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Mecejide (2020). "Coordinates of Oca".