In 7dimensional geometry, a
7simplex
Contents 1 Alternate names 2 As a configuration 3 Coordinates 4 Images 5 Related polytopes 6 Notes 7 External links Alternate names[edit] 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 an octaexon the acronym oca.[1] As a configuration[edit] The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (fvectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[2][3] [ 8 7 21 35 35 21 7 2 28 6 15 20 15 6 3 3 56 5 10 10 5 4 6 4 70 4 6 4 5 10 10 5 56 3 3 6 15 20 15 6 28 2 7 21 35 35 21 7 8 ] displaystyle begin bmatrix begin matrix 8&7&21&35&35&21&7\2&28&6&15&20&15&6\3&3&56&5&10&10&5\4&6&4&70&4&6&4\5&10&10&5&56&3&3\6&15&20&15&6&28&2\7&21&35&35&21&7&8end matrix end bmatrix Coordinates[edit] The Cartesian coordinates of the vertices of an origincentered regular octaexon having edge length 2 are: ( 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , 1 / 3 , ± 1 ) displaystyle left( sqrt 1/28 , sqrt 1/21 , sqrt 1/15 , sqrt 1/10 , sqrt 1/6 , sqrt 1/3 , pm 1right) ( 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , − 2 1 / 3 , 0 ) displaystyle left( sqrt 1/28 , sqrt 1/21 , sqrt 1/15 , sqrt 1/10 , sqrt 1/6 , 2 sqrt 1/3 , 0right) ( 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , − 3 / 2 , 0 , 0 ) displaystyle left( sqrt 1/28 , sqrt 1/21 , sqrt 1/15 , sqrt 1/10 ,  sqrt 3/2 , 0, 0right) ( 1 / 28 , 1 / 21 , 1 / 15 , − 2 2 / 5 , 0 , 0 , 0 ) displaystyle left( sqrt 1/28 , sqrt 1/21 , sqrt 1/15 , 2 sqrt 2/5 , 0, 0, 0right) ( 1 / 28 , 1 / 21 , − 5 / 3 , 0 , 0 , 0 , 0 ) displaystyle left( sqrt 1/28 , sqrt 1/21 ,  sqrt 5/3 , 0, 0, 0, 0right) ( 1 / 28 , − 12 / 7 , 0 , 0 , 0 , 0 , 0 ) displaystyle left( sqrt 1/28 ,  sqrt 12/7 , 0, 0, 0, 0, 0right) ( − 7 / 4 , 0 , 0 , 0 , 0 , 0 , 0 ) displaystyle left( sqrt 7/4 , 0, 0, 0, 0, 0, 0right) More simply, the vertices of the
7simplex
7
Simplex
Model created using straws (edges) and plasticine balls (vertices) in triakis tetrahedral envelope 7
Simplex
7simplex
orthographic projections Ak Coxeter plane A7 A6 A5 Graph Dihedral symmetry [8] [7] [6] Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry [5] [4] [3] Related polytopes[edit] This polytope is a facet in the uniform tessellation 331 with CoxeterDynkin diagram: This polytope is one of 71 uniform 7polytopes with A7 symmetry. A7 polytopes t0 t1 t2 t3 t0,1 t0,2 t1,2 t0,3 t1,3 t2,3 t0,4 t1,4 t2,4 t0,5 t1,5 t0,6 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t1,2,4 t0,3,4 t1,3,4 t2,3,4 t0,1,5 t0,2,5 t1,2,5 t0,3,5 t1,3,5 t0,4,5 t0,1,6 t0,2,6 t0,3,6 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,2,3,4 t1,2,3,4 t0,1,2,5 t0,1,3,5 t0,2,3,5 t1,2,3,5 t0,1,4,5 t0,2,4,5 t1,2,4,5 t0,3,4,5 t0,1,2,6 t0,1,3,6 t0,2,3,6 t0,1,4,6 t0,2,4,6 t0,1,5,6 t0,1,2,3,4 t0,1,2,3,5 t0,1,2,4,5 t0,1,3,4,5 t0,2,3,4,5 t1,2,3,4,5 t0,1,2,3,6 t0,1,2,4,6 t0,1,3,4,6 t0,2,3,4,6 t0,1,2,5,6 t0,1,3,5,6 t0,1,2,3,4,5 t0,1,2,3,4,6 t0,1,2,3,5,6 t0,1,2,4,5,6 t0,1,2,3,4,5,6 Notes[edit] ^ Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o  oca". ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117 External links[edit] Glossary for hyperspace, George Olshevsky. Polytopes of Various Dimensions Multidimensional Glossary v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square pgon Hexagon Pentagon Uniform polyhedron
Tetrahedron
Octahedron
Dodecahedron • Icosahedron Uniform 4polytope
5cell
16cell
Uniform 5polytope
5simplex
5orthoplex
Uniform 6polytope
6simplex
6orthoplex
Uniform 7polytope
7simplex
7orthoplex
Uniform 8polytope
8simplex
8orthoplex
Uniform 9polytope
9simplex
9orthoplex
Uniform 10polytope
10simplex
10orthoplex
Uniform npolytope nsimplex northoplex • ncube ndemicube 1k2 • 2k1 • k21 npentagonal polytope Topics:
Polytope
