Fractal Fourier Series

Faculty Member

Prof. A J Hildebrand

Project Image

Fourier Series with fractal-like properties arise in a wide range of
applications, from calculus to analysis, fractal geometry, number theory, thermodynamics, and turbulence. Despite their rather simple definition via a convergent infinite series, the graphs of such series can exhibit surprisingly complex behavior. In some cases the graphs appear to be completely chaotic and unpredictable, while in other cases they have very distinctive spiral-like features or show fractal behavior. Small changes in a single parameter can cause dramatic changes in the appearance graph. Much of this behavior is still very poorly understood.

In this project we will explore fractal Fourier series from an experimental point of view, using Mathematica as the primary coding tool. The main goals of the project are to (1) develop Mathematica code to facilitate systematic and large scale experimentation with such series; (2) use such experimentation to unravel some of the mysteries behind these series and explain the observed behavior; and (3) create an interactive Mathematica-based visualization for publication at the Wolfram Demonstrations website, http://www.demonstrations.wolfram.com.

This project is similar in spirit to (though independent from) a Fall
2021 IGL project titled "Interactive Visualizations with Mathematica". For more about the latter project see the following links:

https://wiki.illinois.edu/wiki/display/IGL/Interactive+Visualizations+with+Mathematica

https://demonstrations.wolfram.com/FractalCurvesGeneratedByFourierSeries/
 

Project Difficulty

Intermediate

Undergrad Prerequisites

Aside from completion of the calculus sequence, there are no specific course work requirements. In particular, no prior knowledge of Fourier series or fractals is expected.

Experience with Mathematica, or evidence of very strong general coding skills and ability to quickly pick a new programming language, is essential.