Fourier, Gauss, Fraunhofer, Porod and the shape from moments problem

We show how the Fourier transform of a shape in any number of dimensions can be simplified using Gauss's law and evaluated explicitly for polygons in two dimensions, polyhedra in three dimensions, etc. We also show how this combination of Fourier and Gauss can be related to numerous classical problems in physics and mathematics. Examples include Fraunhofer diffraction patterns, Porod's law, the shape from moments problem, and Davis's extension of the Motzkin-Schoenberg formula to polygons in the complex plane