Algebraic Starscapes

The Mathematics

Fixing a degree, every integer polynomial of that degree is a single point in a lattice of coefficients, carrying with it a set of complex roots. Plotting those roots — colored by degree — reveals how the two are related. The black points are the roots of integer quadratics: the subset of the coefficient lattice ℤ³ with negative discriminant, drawn in the upper half-plane, where they trace a fractal woven from the geodesics of the SL(2,ℤ) tiling of the hyperbolic plane. The red points are the roots of integer cubics, and the purple points those of some quartics — patterns whose precise geometry we have yet to understand.

Technique

For each degree (2, 3, 4) we build a large N-cube of integer polynomials and discard those with only real roots. The roots in the upper half-plane of everything that remains form a vast point cloud, which we render as disks whose radius is proportional to the polynomial’s discriminant — measured in the hyperbolic metric on the upper half-plane.