Regular octaexon (7simplex) 


Orthogonal projection inside Petrie polygon 

Type  Regular 7polytope 
Family  simplex 
6faces  8 6simplex 
5faces  28 5simplex 
4faces  56 5cell 
Cells  70 tetrahedron 
Faces  56 triangle 
Edges  28 
Vertices  8 
Vertex figure  6simplex 
Petrie polygon  octagon 
Schläfli symbol  {3,3,3,3,3,3} 
CoxeterDynkin diagram  
Coxeter group  A_{7} [3,3,3,3,3,3] 
Dual  Selfdual 
Properties  convex 
In geometry, an octexon, or octa7tope is a 7simplex, a selfdual regular 7polytope with 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5cell 5faces, 28 5simplex 6faces, and 8 6simplex 7faces.
The name octaexon is derived from octa for eight facets in Greek and ex for having sixdimensional facets, and on.
The Cartesian coordinates of the vertices of an origincentered regular octaexon having edge length 2 are:
Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
Regular octaexon (7simplex)  

Orthogonal projection inside Petrie polygon  
Type  Regular 7polytope 
Family  simplex 
6faces  8 6simplex 
5faces  28 5simplex 
4faces  56 5cell 
Cells  70 tetrahedron 
Faces  56 triangle 
Edges  28 
Vertices  8 
Vertex figure  6simplex 
Petrie polygon  octagon 
Schläfli symbol  {3,3,3,3,3,3} 
CoxeterDynkin diagram  
Coxeter group  A_{7} [3,3,3,3,3,3] 
Dual  Selfdual 
Properties  convex 
In 7dimensional geometry, a 7simplex is a selfdual regular 7polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5cell 5faces, 28 5simplex 6faces, and 8 6simplex 7faces. Its dihedral angle is cos^{−1}(1/7), or approximately 81.79°.
Contents 
It can also be called an octaexon, or octa7tope, as an 8facetted polytope in 7dimensions. The name octaexon is derived from octa for eight facets in Greek and ex for having sixdimensional facets, and on. Jonathan Bowers gives a octaexon the acronym oca.^{[1]}
The Cartesian coordinates of the vertices of an origincentered regular octaexon having edge length 2 are:
More simply, the vertices of the 7simplex can be positioned in 8space as permutations of (0,0,0,0,0,0,0,1). This construction is based on facets of the 8orthoplex.
A_{k} Coxeter plane  A_{7}  A_{6}  A_{5} 

Graph  
Dihedral symmetry  [8]  [7]  [6] 
A_{k} Coxeter plane  A_{4}  A_{3}  A_{2} 
Graph  
Dihedral symmetry  [5]  [4]  [3] 
This polytope is one of 71 uniform 7polytopes with A_{7} symmetry.
Fundamental convex regular and uniform polytopes in dimensions 2–10  

n  nSimplex  nOrthoplex • nHypercube  nDemicube  1_{k2} • 2_{k1} • k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{n}  GH_{n}  
Regular polygon  Triangle  Square  Square  Pentagon  
Uniform polyhedron  Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
Uniform polychoron  5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5orthoplex • 5cube  5demicube  1_{12} • 2_{11}  
Uniform 6polytope  6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
Uniform 7polytope  7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
Uniform 8polytope  8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
Uniform 9polytope  9simplex  9orthoplex • 9cube  9demicube  
Uniform 10polytope  10simplex  10orthoplex • 10cube  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
