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:
|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 one of 71 uniform 7-polytopes with A7 symmetry.
- Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o - oca".
- Coxeter, Regular Polytopes, sec 1.8 Configurations
- Coxeter, Complex Regular Polytopes, p.117
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
|An||Bn||I2(p) / Dn||E6 / E7 / E8 / F4 / G2||Hn|
|Tetrahedron||Octahedron • Cube||Demicube||Dodecahedron • Icosahedron|
|5-cell||16-cell • Tesseract||Demitesseract||24-cell||120-cell • 600-cell|
|5-simplex||5-orthoplex • 5-cube||5-demicube|
|6-simplex||6-orthoplex • 6-cube||6-demicube||122 • 221|
|7-simplex||7-orthoplex • 7-cube||7-demicube||132 • 231 • 321|
|8-simplex||8-orthoplex • 8-cube||8-demicube||142 • 241 • 421|
|9-simplex||9-orthoplex • 9-cube||9-demicube|
|10-simplex||10-orthoplex • 10-cube||10-demicube|
|n-simplex||n-orthoplex • n-cube||n-demicube||1k2 • 2k1 • k21||n-pentagonal polytope|
|Topics: Polytope families • Regular polytope • List of regular polytopes and compounds|