  In analogy with the Poisson summation formula, we identify when the
fractional Fourier transform, applied to a Dirac comb in dimension one, gives a
discretely supported measure. We describe the resulting series of complex
multiples of delta functions, and through either the metaplectic representation
of $SL(2,\mathbb{Z})$ or the Bargmann transform, we see that the the
identification of these measures is equivalent to the functional equation for
the Jacobi theta functions. In tracing the values of the antiderivative in
certain small-angle limits, we observe Euler spirals, and on a smaller scale,
these spirals are made up of Gauss sums which give the coefficient in the
aforementioned functional equation.
