A Gallery of Gaussian Periods

TL;DR

This paper introduces Gaussian periods, showcases their visual patterns through new plots generated by an improved app, and explores the mathematical principles behind their visual features, expanding understanding of these classical sums.

Contribution

The paper presents a new app for efficiently creating large-scale Gaussian period plots and analyzes the mathematical factors influencing their visual patterns.

Findings

Gaussian periods exhibit striking visual patterns.

Mathematical properties influence colorings and patterns.

New visualization tools enable large-scale exploration.

Abstract

Gaussian periods are certain sums of roots of unity whose study dates back to Gauss's seminal work in algebra and number theory. Recently, large scale plots of Gaussian periods have been revealed to exhibit striking visual patterns, some of which have been explored in the second named author's prior work. In 2020, the first named author produced a new app, \texttt{Gaussian periods}, which allows anyone to create these plots much more efficiently and at a larger scale than before. In this paper, we introduce Gaussian periods, present illustrations created with the new app, and summarize how mathematics controls some visual features, including colorings left unexplained in earlier work.