An Exploration of Dimension

What do you think of when you hear the word "dimension"? Maybe you think of the fact that we live in 3 dimensional space, or how much you love/hate 3D movies, or the dimension hopping hero from your favorite sci-fi show. Regardless, we all have an intuitive idea of what "dimension" means. Points are in dimension 0, lines are in dimension 1, flat disks in dimension 2, solid balls in dimension 3, and so on. Sure things get harder to picture in dimension 4 and up, but wait it looks like we skipped a bunch of numbers when talking about dimensions. What about dimension 1/2 or log(3)/log(2)? Does dimension log(3)/log(2) even make sense?... It will if you choose to join our exploration.

In this project we will explore different notions of dimension including the intuitive topological dimension and the stranger Hausdorff dimension. We'll also talk about the kinds of weirdo sets that "live well" in dimensions like log(3)/log(2), the same way a disk "lives well" in dimension 2. These wacko sets are often called fractals and, time permitting, we'll see why it's so difficult to define precisely what the word "fractal" means.