4 21 Polytope

4 21 Polytope

In 8-dimensional geometry, the 4₂₁ is a semiregular uniform 8-polytope, constructed within the symmetry of the E₈ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

Coxeter named it 4₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, .

The rectified 4₂₁ is constructed by points at the mid-edges of the 4₂₁. The birectified 4₂₁ is constructed by points at the triangle face centers of the 4₂₁. The trirectified 4₂₁ is constructed by points at the tetrahedral centers of the 4₂₁, and is the same as the rectified 1₄₂.

These polytopes are part of a family of 255 = 28 − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: .