Jan Philipp Birmanns

An epicycle is typically understood as a rotating vector – a type of mathematical arrow – that is connected to another rotating vector. When many such arrows are linked together, one can manipulate the movement of the final arrow by adjusting the length and frequency of rotation of each epicycle before it. Therefore, one can use epicycles to create any drawing imaginable. Selecting the right arrows for a specific result can be tedious. This issue can be solved by running different calculations that utilize the Fourier Transform, an equation that will break down periodic signals into sets of sine and cosine waves. Thanks to modern technology, the Fourier Transform is one of the most-used equations in the world; it plays a vital role in most forms of wireless transmission and many methods of data compression. Within this project, I focused on utilizing the Fourier Transform for drawing with epicycles. I first found proofs and explanations for existing strategies, as well as possible improvements. I then generalized these findings for four-dimensional drawings, showing that the phenomenon is not limited to the plane. One can imagine a four-dimensional point as a position in three-dimensional space that has a color. When every color has been assigned a value, the point can represent four separate numbers. Nonetheless, the behavior of the generalized strategy was also investigated for three-dimensional drawings. To better communicate these findings, I developed the websites dft.birmanns.org and dqft.birmanns.org. Here users can harness the power of epicycles to recreate their own drawings.