I introduce Kolmogorov complexity and the Erdos distinct distance problem, describe an intuitive connection between these topics, and explore whether they are actually related. I construct sets in \(\mathbb{R}^2\) with arbitrary Kolmogorov complexity and distance set size in order to show that the Kolmogorov complexity and the size of the distance set for finite sets of fixed size are independent quantities. I consider what alternative descriptions of set complexity might agree more closely with my geometric intuition.