Regular octaexon (7-simplex) | |
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Orthogonal projection inside Petrie polygon | |
Type | Regular 7-polytope |
Family | simplex |
Schläfli symbol | {3,3,3,3,3,3} |
Coxeter-Dynkin diagram | |
6-faces | 8 6-simplex |
5-faces | 28 5-simplex |
4-faces | 56 5-cell |
Cells | 70 tetrahedron |
Faces | 56 triangle |
Edges | 28 |
Vertices | 8 |
Vertex figure | 6-simplex |
Petrie polygon | octagon |
Coxeter group | A_{7} [3,3,3,3,3,3] |
Dual | Self-dual |
Properties | convex |
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.^{[1]}
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.^{[2]}^{[3]}
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 |
A_{k}Coxeter plane | A_{7} | A_{6} | A_{5} |
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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 a facet in the uniform tessellation 3_{31} with Coxeter-Dynkin diagram:
This polytope is one of 71 uniform 7-polytopes with A_{7} symmetry.