Mathematik  |  Informatik

Jan Birmanns, 2003 | Rheinau, ZH

Epicycles offer a fascinating method for creating drawings in two-, three-, and four-dimensional spaces. The Fourier Transform, a critical component of modern technology, lies at the heart of this process. It enables the construction of a set of interpolation methods that may be interpreted as sequences of epicycles or rotating arrows. To showcase their striking behavior to readers, two pieces of software have been developed. They can be accessed at dft.birmanns.org and dqft.birmanns.org respectively, allowing users to draw with epicycles. In the context of this project, rigorous proofs were found to explain the phenomenon. This information served as a foundation for several improvements that could be applied to the original methods. To introduce readers to these findings, they will also be familiarized with the underlying mathematical groundwork, including complex numbers and quaternions.

Introduction

This paper explores the question of how one can use epicycles to interpolate in two-, three-, and four-dimensional spaces. Epicycles were first used to describe planetary motion in ancient times. The term may refer to a circle moving along the circumference of another circle or a rotating arrow. The websites dft.birmanns.org and dqft.birmanns.org provide users with the ability to create two- or three-dimensional drawings that are recreated by epicycles.

Methods

By constructing a precise mathematical foundation, the phenomenon could be discussed in a concise and formal manner. It relies on an interpolation method created by combining the Discrete Fourier Transform with its inverse. An alternate interpretation allows it to appear as various epicycles chained together. Precise analysis of its structure allowed for various attempts at improving its efficiency and the appearance of the resulting images. As this procedure was limited to points located on the complex plane, the quaternion space was also explored. A similar method for up to four-dimensional values could be established, relying on the Discrete Quaternion Fourier Transform. The possibility of interpreting it too as a chain of epicycles was studied in much detail.

Results

Great success was achieved in improving the Discrete Fourier Transform for this application. Depending on the input values, it can be impossible to differentiate between the original drawings and ones created with epicycles. Efficiency could be improved by implementing the Fast Fourier Transform. These changes made creating a piece of software possible that can draw with epicycles on a plane. It was shown that the same could be done for the Discrete Quaternion Fourier Transform in quaternion space. However, some adjustments had to be made for three dimensions and elliptical epicycles were introduced. Although a Fast Quaternion Fourier Transform was introduced, it requires significantly more computational power. Once again, a website was created where different points in space can be picked and traced by elliptical epicycles. The animations it generates are intriguing even to users unfamiliar with the topic.

Discussion

This project provides precise answers to the question it set out to explore. The novel goal of finding the most optimal strategy was also pursued. However, it fell short in finding an efficient strategy for epicycles in three- and four-dimensional spaces. Further, no general procedure was uncovered to draw with epicycles in three dimensions without elliptical epicycles. Its nonexistence could only be proven for interpolation methods based on the Discrete Fourier Transform. Nonetheless, this approach was well suited for the complex plane and quaternion space. Representation of the latter posed difficulties until the fourth dimension was visualized through color.

Conclusions

There is still much left to be uncovered in the context of Fourier Analysis and epicycles. In addition to improving efficiency, further exploration of its potential in spaces with more than four dimensions could prove valuable. It might also be beneficial to research alternate strategies that do not require elliptical epicycles in three-dimensional space. An increase in efficiency, on the other hand, could benefit the demonstrations at dft.birmanns.org and dqft.birmanns.org.

Würdigung durch den Experten

Prof. Dr. Norbert Hungerbühler

Wie kann man eine vorgegebene Kurve im n-dimensionalen euklidischen Raum durch eine mathematische Formel beschreiben und nachzeichnen? Dies ist eine natürliche Frage von hoher Praxisrelevanz. Jan Birmanns zeigt eine elegante Lösung mithilfe von Fourier-Reihen. In der Ebene bedient er sich der diskreten Fourier-Transformation, im drei- und vierdimensionalen Raum gelingt ihm ein besonders effizienter Zugang durch die diskrete Quaternionen-Fourier-Transformation. Im Rahmen der Arbeit entstand auch eine Softwarelösung, die es dem Benutzer erlaubt, Kurven vorzugeben, die dann nachgezeichnet werden.

Prädikat:

hervorragend

Sonderpreis «Stockholm International Youth Science Seminar (SIYSS)» gestiftet von der Metrohm Stiftung

Kantonsschule Im Lee, Winterthur
Lehrerin: Nicoletta Ravizza-Andri