In 8-dimensional geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the 4₂₁ is a semiregular uniform 8-polytope, constructed within the symmetry of the E₈

E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

 group. It was discovered by Thorold Gosset

Thorold Gosset

Thorold Gosset was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher.According to H. S. M...

, published in his 1900 paper. He called it an 8-ic semi-regular figure.

Coxeter named it 4₂₁ by its bifurcating Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, 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.