Project 3H 2022/23 Durham University

We know intuitively that the dimension of a line segment is 1 and the dimension of a solid disk is 2. The Sierpinski gasket is the complicated set indicted in the figure. It contains a complicated collection of lines, so surely is not 1 dimensional, but seems not solid enough to be 2 dimensional. What should its dimension be? The consensus is log 3 / log 2. Notice that the image is made up of 3 copies of itself with scaling factor 1/2.

The Sierpinski gasket is an example of what we call a fractal set. There is a mathematically rigorous way to define the dimension of any set. (Actually there a few different kinds of dimension.) This works especially well for sets with a nice kind of fractal structure.

The aim of this project is to understand the notion of dimension and our abilities and limitations in computing it.

Pre-requisites: Complex Analysis 2.

Reference: Fractal Geometry: Mathematical Foundations and Applications, K. J. Falconer.