Abstract: A space-filling curve is a continuous map from the unit interval onto the unit square. The graphs of coordinate functions of space-filling curves such as those described by Peano, Hilbert, Polya and others, are typical examples of self-affine sets.

The first half of this talk, I describe how the study of dimensions of self-affine sets was motivated, at least in part, by these coordinate functions and their natural generalizations, and review the relevant literature. In the second part, I present our results (with Pieter Allaart) on the coordinate functions of Levy's dragon curve, which is not a space-filling curve but certainly a strange curve with its dimension 2. Finally, I will give two concrete open problems. They are accessible for undergraduate/graduate students who are looking for interesting math problems.

This talk will be illustrated with many beautiful pictures including 3-dimensional animation assisted by Mike Trenfield, (UNT senior undergraduate student).