How Accurate Does Pi Need to Be?

I just want to make a little clearer what I was just writing about when I was speaking about a quantum mechanical limit to the value for pi. The calculation for pi assumes a purely Euclidean geometry. The real world seems to follow the geometry of General Relativity (Riemannian geometry). Euclidean geometry works as a good approximation in low gravity for short distances.  That means at some point quantum mechanics and general relativity conspire to mean that pi needs only be accurate up to that point.

Some physicist who is up to the math should look at this It probably turns out to coincide with a known  phenomenon. Perhaps someone else has already done the work and I just haven't run into it yet.