Cubical Covers of Sets in R^n

In this paper, we explore several methods of representing subsets of R n using their geometric and analytic properties. We present a heuristic, expository approach to estimating the size of various sets and their boundaries, with the goal of preserving important features in the representations. The aim is to stimulate interest in and serve as an introduction to this topic as well as two other well known methods: Jones' numbers and varifolds from geometric measure theory. We provide various computations and exercises, numerous illustrations, suggestions for further research, and an extensive list of current resources and references in these areas.