We introduce the topic of the Fourier transform of a Euclidean polytope, first by examples and then by more general formulations. Then we point out how we can use this transform (and the frequency space) to analyze the following problems:

1. Compute lattice point enumeration formulas for polytopes

2. Relate the transforms of polytopes to tilings of Euclidean space by translations of a polytope

We will give a flavor of how such applications arise, and we point to some conjectures and applications.