Suppose that $\mathcal{R}$ is a first order expansion of $( \mathbb{R}, <, +, ( x \mapsto \lambda x)_{\lambda \in \mathbb{R}})$. It has recently become apparent that a general dichotomy holds: either every closed $\mathcal{R}$-definable set enjoys certain smoothness properties or every compact set is $\mathcal{R}$-definable. In more geometric language: if $X \subseteq \mathbb{R}^k$ is closed and "highly singular" or "fractal" then every compact subset of every $\mathbb{R}^n$ can be constructed from $X$ using finitely many boolean operations, cartesian products, and linear operations. The development of this topic requires the development of o-minimal tools in a maximally general setting.