  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 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, Porods law,
Hopfs Umlaufsatz, the isoperimetric inequality and Didos problem. We also use
this approach to provide an alternative derivation of Davis's extension of the
Motzkin-Schoenberg formula to polygons in the complex plane.
