Abstract: In the study of analysis on metric spaces, rectifiable sets are the appropriate analogue of smooth manifolds. Due to a celebrated theorem of Rademacher, we know rectifiable sets have a "sort of" almost-everywhere-differentiable structure with respect to which one can define approximate tangent planes and do calculus at almost every point. Classifying rectifiable or, on the other end of the spectrum, purely unrectifiable sets is not an easy task. In this talk we will see a certain classification of purely unrectifiable sets via orthogonal projections, will see how Fourier analysis plays a key role in this subject and will talk about how (if at all) similar classification can be apply to more general metric spaces, specifically the Heisenberg Group. Note: This talk will be expository, introductory and for the most part self-contained. No knowledge beyond a basic understanding of metric spaces and measure theory will be needed to follow along.