The distinct symmetries of the closed unit interval [0,1] are relatively easy to understand. The problem becomes much more complex if one is to consider the closed unit square [0,1]^2 instead. Indeed it is a theorem of Hjorth that it is not possible to definably classify the symmetries of the square by using reasonably simple invariants.

In this talk, we will rigorously define terms such as "distinct symmetries" and "reasonably simple invariants." We also briefly discuss Hjorth's theory of turbulence used to prove the above result. Finally, we take a step towards studying the more general interplay between dimension and turbulence by arguing that even the distinct symmetries of the Sierpinski carpet, which will also be defined, cannot be classified in this manner.