Consider a sphere of size on the order of the size of the observable universe. Suppose we wish to compute its volume to the precision of a Planck volume (one cubic Planck length). How many digits of pi are needed?
The question is complicated because we do not assume the universe is flat, a Euclidean manifold. To what precision do we currently know the curvature of the universe? That is, beyond how many digits of pi does the uncertainty of the curvature parameter drown out further precision in pi? (There's also uncertainty in Planck length.)
What is the formula for volume of a sphere in a space with a given Gaussian curvature?
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