7-simplex

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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 hexatera
Tera56 pentachora
Cells70 tetrahedra
Faces56 triangles
Edges28
Vertices8
Vertex figure6-simplex, edge length 1
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Cell radius
Teron radius
Peton radius
Inradius
Hypervolume
Diexal angle
HeightsPoint atop hop:
 Dyad atop perp hix:
 Trig atop perp pen:
 Tet atop perp tet:
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
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[edit | edit source]

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

  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • .

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

  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • .

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

  • .

Representations[edit | edit source]

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[edit | edit source]

  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".

External links[edit | edit source]