2 21 Polytope

2 21 Polytope

In 6-dimensional geometry, the 2₂₁ polytope is a uniform 6-polytope, constructed within the symmetry of the E₆ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure.

Coxeter named it 2₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 2₂₁.

The rectified 2₂₁ is constructed by points at the mid-edges of the 2₂₁. The birectified 2₂₁ is constructed by points at the triangle face centers of the 2₂₁, and is the same as the rectified 1₂₂.

These polytopes are a part of family of 39 convex uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .