Gabriel Dorfsman-Hopkins, Eliza Brown

Statement

The fundamental theorem of algebra states that every polynomial equation with complex coefficients has complex roots. When families of these roots are plotted on the complex plane, we obtain striking patterns called algebraic starscapes. In this series, we created starscapes from the eigenvalues of all 3x3 integer matrices within fixed bounds, sizing and coloring the roots according to the determinant. We developed python code to generate the images—utilizing techniques such as kernel cropping and gaussian blurring—and employed a supercomputer to parallelize the millions of computations. The produced images exhibit deep relationships between the geometry and arithmetic and serve as invitations to explore the mysterious patterns of integers.

Artworks

This starscape depicts the complex eigenvalues of all 3x3 matrices with integer entries between -6 and 0. The size and brightness of each point are inversely proportional to the determinant of the generating matrix. There are over 120 million such eigenvalues (counted with multiplicity), so shrinking points with larger determinant allows sharp visual patterns to emerge. These patterns reflect structures across different fields of math, including the theory of linear operators, algebraic number theory, as well as the hexagonal projection of the geometry of the 9 dimensional cube that we sample from. We hope the viewer is enchanted by these mysterious structures, as mathematicians have been for centuries.

This starscape depicts the complex eigenvalues of all 3x3 matrices with integer entries between -3 and 3. The size and brightness of each point are inversely proportional to the determinant of the generating matrix, and the color of the points are reflect the determinant of the generating matrix (with blue points having positive determinant, and red points having negative determinant). The patterns reflect structures across different fields of math, including the theory of linear operators, algebraic number theory. A few surprises emerge, such as a tendency for eigenvalues with positive real parts coming from matrices with negative determinants. We hope the viewer, like a mathematician, is drawn in by the beauty to demand explanations.