This book chapter is published open access.
In mathematics, we are often drawn to the simple or elegant, but what lies at the other end of the spectrum? How can we build and study complex objects? How can we break them down? In this note, we will describe some tools for building functions and surfaces with structure at many different scales and, conversely, tools for decomposing complex objects into simple pieces. These methods are based on ideas from geometric measure theory and harmonic analysis, and we will give some applications to quantitative and metric geometry.