PROJECT III 2021-22

FRACTALS, SUBSTITUTION TILINGS AND LINDENMAYER SYSTEMS

This project touches on three areas of Mathematics which all share the common idea that rich and often beautiful complexity can be generated by iterating a small collection of simple rules. The project can be pursued in any number of ways, for example by studying one topic individually, by looking at the links between two or more of them, or even by trying to develop new connections.

Fractal objects and images are well known even among non-mathematical audiences through famous pictures such as those of the Mandelbrot set. They typically have properties of self-similarity under magnification.

Substitution tilings are a class of decorations of the plane (and other dimensions of space) which have the inverse sort of self-similarity: self-similarity under “microscopification”. Again, some examples such as the Penrose tiling, are well known to the wider world.

Lindenmayer systems were developed as a simple tool to model complex biological systems via repeated operations of basic rules at varying scales. Originally introduced as a theoretical framework for studying the development of multicellular organisms, they are also frequently used to generate realistic computer images of the natural environment.

PREREQUISITES. No specific prerequisites other than the core (mainly pure) maths background of Levels I and II. Students who wish to use these ideas to develop images would need some programming background, but the project certainly does not need to be taken in that direction.

RESOUCES. There are many good introductions to these topics to be found on the web. For some specific, and easily available, suggestions, you could take a look at the following

• K. J. Falconer Fractals: a very short introduction, OUP 2013 (available electronically at the University library); see also the same author’s larger book Fractal Geometry (especially Part II Applications and Examples)

• N. Frank, A primer of substitution tilings of the Euclidean plane, Expo. Math. 26 (2008), no. 4, 295-326. Also available via her webpage https://pages.vassar.edu/nafrank/?page_id=15

• And for an overview of Lindenmayer systems and some links to fractals and tilings, you could start with the wikipedia page for L-Systems, https://en.wikipedia.org/wiki/L-system

EMAIL

John Hunton (mailto:john.hunton@durham.ac.uk)