Regular enneazetton (8-simplex) | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 8-polytope |
Family | simplex |
Schläfli symbol | {3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | |
7-faces | 9 7-simplex |
6-faces | 36 6-simplex |
5-faces | 84 5-simplex |
4-faces | 126 5-cell |
Cells | 126 tetrahedron |
Faces | 84 triangle |
Edges | 36 |
Vertices | 9 |
Vertex figure | 7-simplex |
Petrie polygon | enneagon |
Coxeter group | A_{8} [3,3,3,3,3,3,3] |
Dual | Self-dual |
Properties | convex |
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos^{−1}(1/8), or approximately 82.82°.
It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on.
This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-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.^{[1]}^{[2]}
The Cartesian coordinates of the vertices of an origin-centered regular enneazetton having edge length 2 are:
More simply, the vertices of the 8-simplex can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on facets of the 9-orthoplex.
Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.
A_{k} Coxeter plane | A_{8} | A_{7} | A_{6} | A_{5} |
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Graph | ||||
Dihedral symmetry | [9] | [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 tessellations: 2_{51}, and 5_{21} with respective Coxeter-Dynkin diagrams:
This polytope is one of 135 uniform 8-polytopes with A_{8} symmetry.
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