3 21 Polytope

3 21 Polytope

In 7-dimensional geometry, the 3₂₁ polytope is a uniform 7-polytope, constructed within the symmetry of the E₇ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

Coxeter named it 3₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.

The rectified 3₂₁ is constructed by points at the mid-edges of the 3₂₁. The birectified 3₂₁ is constructed by points at the triangle face centers of the 3₂₁. The trirectified 3₂₁ is constructed by points at the tetrahedral centers of the 3₂₁, and is the same as the rectified 1₃₂.

These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .