Apologies if this is the wrong subreddit, I am new to Reddit.
As a math lover, I have always appreciated a specific link between the Fourier transform and the Normal distribution that I wanted to share. I am probably biased because I worked with Fourier transforms and Monte Carlo methods in the past, but I still find this connection incredibly elegant.
For a normalized Fourier transform F, the operator has four eigenvalues (since F\^4 = I): 1, i, -1, and -i. When you work out the corresponding eigenspaces, it turns out that you can generate each eigenspace using the Hermite Functions (which are built from Hermite polynomials multiplied by a Gaussian weight).
Specifically, the zeroth-degree Hermite function is exactly the density function of the standard normal distribution (a Gaussian). Because its eigenvalue is 1, it is a fixed point of the transform. This provides a geometric way to see why the normal distribution is a fixed point of the characteristic function (which is essentially the Fourier transform of the probability density).
For me, it just feels like everything is made of Gaussian distributions.