6-simplex | |
---|---|
Type | uniform polypeton |
Schläfli symbol | {3^{5}} |
Coxeter diagrams | |
Elements |
f_{5} = 7, f_{4} = 21, C = 35, F = 35, E = 21, V = 7 |
Coxeter group | A_{6}, [3^{5}], order 5040 |
Bowers name and (acronym) |
Heptapeton (hop) |
Vertex figure | 5-simplex |
Circumradius | 0.654654^{[1]} |
Properties | convex, isogonal self-dual |
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos^{−1}(1/6), or approximately 80.41°.
It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop.^{[2]}
This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-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.^{[3]}^{[4]}
The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:
The vertices of the 6-simplex can be more simply positioned in 7-space as permutations of:
This construction is based on facets of the 7-orthoplex.
A_{k} Coxeter plane | A_{6} | A_{5} | A_{4} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
A_{k} Coxeter plane | A_{3} | A_{2} | |
Graph | |||
Dihedral symmetry | [4] | [3] |
The regular 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A_{6} Coxeter plane orthographic projections.