There isn't a way to derive $\pi$ because it's a fundamental constant and not something that can be derived.
However there are lots and lots of ways to approximate $\pi$. I believe that Archimedes was the first person to write down such an approximation (in 250BC), and we've been developing better and better ways of approximating $\pi$ since.
One of the approximations for $\pi$ is due to a mathematician called John Wallis in 1655 and is called the Wallis product. There's nothing special about his approximation - it's just one of the many that have been discovered. The recent paper reports that the Wallis product has turned up in some calculations about hydrogen atoms.
This sort of thing amuses both physicists and mathematicians because it's always fun when some unexpected link is made between physics and maths. However there is nothing in this that will revolutionise our understanding of physics. These sorts of links are pretty frequent and are discovered on a regular basis.