# Hausdorff Dimension

## Lisa Naples (University of Connecticut)

### Wednesday, October 25, 2017 5:45 pmMONT 314

The size of simple shapes are measured in different ways depending on their dimension. For 1-dimensional shapes, like a line or curve, we use length. For 2-dimensional shapes, like rectangles and triangles, we use area. For 3-dimensional shapes, like a solid cube or cylinder, we use volume.

In calculus we compute length, area, and volume using limits of Riemann sums: ordinary integrals, double integrals, and triple integrals. Each type of integral depends on the dimension of the shape. It turns out there are some shapes that are so wild their dimension is no longer a number like 1, 2, or 3 but something in between. How do we determine the dimension of such a set and what is a good measure of its size adapted to its dimension?

In this talk we will meet the Cantor set, a classical example of a fractal, and discuss one way to show its dimension is $\log_3 2$ and measure its size using a method designed for shapes with that dimension. We also discuss how we can apply the techniques in this example to other fractals.