David Lowry-Duda

This composition features the 19 nontrivial Dirichlet characters mod 33. (The trivial Dirichlet character would consist of a single point). We typically think of Dirichlet characters as discrete group homomorphisms from a finite group, but in analytic number theory it is often convenient to interpolate the character to a map from the real numbers. Identifying these characters with their interpolations and plotting the images of the real line yields the collection of images here. Each character takes values on the unit circle and at the origin, which is why we can recognize underlying circles. Some of the characters are unique, while others form sets that transform into each other through a symmetry (and which lie in the same Galois orbit).

This composition is based on visualizing rational points on the unit circle. The challenge in visualizing these rational points is that they all lie on the circumference, and that hides the underlying structure. Here, each point is plotted at the correct angle, but at a radius equal to the largest denominator (what an arithmetic geometer might call naive height) of the coordinates of that point. Close examination shows the 4 points (1,0), (0, 1), (-1, 0), and (0, -1) near the center, and then four notable rays without rational points in the cardinal directions. Trying to understand this avoidance first led me towards this picture. Countless parabolas manifest among the points, hinting at height-preserving projections to conics.