Gallery of Limit roots

Geometry of infinite root systems

Here is a gallery of limit roots of infinite Coxeter groups produced by Sage.

Normalized isotropic cone and the first few thousand normalized roots for the Coxeter graph depicted. The set of accumulation points is dense on the isotropic cone.

Normalized isotropic cone and the first few thousand normalized roots for the universal Coxeter group on 4 generator. The set of accumulation points form an Apollonian gasket.

These pictures are artistic variations with different colors and size for the normalized roots (the closer the root to the isotropic cone, the smaller it is represented).