The Pi Manifesto cited the Gaussian integral as a win for pi and claimed that mathematicians would agree that the pi version is more natural, but when we actually look into Gaussian functions, we will see just how specious this argument is.
The definition of a Gaussian function is a function of the form: https://wikimedia.org/api/rest_v1/media ... be14d641f1
Notice how the factor of 1/2 in the exponential is part of the definition of a Gaussian function, so there is nothing unnatural about having a factor of 1/2 in the power of e in the tau version; in fact, it is unnatural to not have it in the pi version. The tau version of the Gaussian integral comes from setting a=1, b=0, and c=1. The pi version comes from setting a=1, b=0, and c=sqrt(2)/2. Which one of those sounds more natural?
The only reason pi shows up there is because c^2=1/2. However, the integral of ANY Gaussian is a times the absolute value of c times root tau. Because of this, the integral is only equal to one when a=1/(c√τ). This is where the factor of 1/σ√τ in the normal distribution comes from. It has nothing to do with the factor of 1/2 in the exponential, so grouping the σ with √2 like the Pi Manifesto did is really just a silly attempt to make it look like pi is natural there with no actual logical backup. The normal distribution is still a win for tau.
Now to go off on a bit of a sin/cos, there is also a two-dimensional Gaussian function, and its integral is https://wikimedia.org/api/rest_v1/media ... 9280696c7a