inside Petrie polygon
|Coxeter group||A7 [3,3,3,3,3,3]|
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°.
It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca.
As a configurationEdit
This configuration matrix represents the 7-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.
The Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are:
More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of (0,0,0,0,0,0,0,1). This construction is based on facets of the 8-orthoplex.
|7-Simplex in 3D|
Ball and stick model in triakis tetrahedral envelope
7-Simplex as an Amplituhedron Surface
7-simplex to 3D with camera perspective showing hints of its 2D Petrie projection
|Ak Coxeter plane||A7||A6||A5|
|Ak Coxeter plane||A4||A3||A2|
This polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram:
This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
- ^ Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o3o — oca".
- ^ Coxeter, H.S.M. (1973). "§1.8 Configurations". Regular Polytopes (3rd ed.). Dover. ISBN 0-486-61480-8.
- ^ Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary