Powers of Pi and Unsolved problems.

Steven Strogatz tweeted about a really interesting paper yesterday:

http://www.americanscientist.org/libraries/documents/201110101628308738-2011-11CompSciHayes.pdf

The paper discusses a pretty simple question – what is the volume of a unit sphere in n dimensions?  The answer has at least one really cool surprise, and the whole paper is a great read.

I can’t remember ever thinking about these volumes before and was pleasantly surprised to see that these spherical volumes had increasing powers of  $\pi$ in their formulas.   That result reminded me of another famous problem whose solution also has powers of $\pi$ – summing inverse powers of integers.

In the 1700s Euler found a closed form for the sum of the reciprocals of the integers raised to any positive even integer.  For example:

$\sum_{1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}$ and $\sum_{1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}.$

He was unable to find a sum for any of the odd powers, though, and that problem remains unsolved even today.    Finding a closed form for the some of the inverse cubes is a problem that absolutely fascinated me as a kid, and I guess it remains in the back of my mind even today.  My assumption was  that there was some undiscovered connection between geometry and this sum that would provide an answer to the problem.  Before seeing the paper about the volumes of higher dimensional spheres, I don’t remember ever previously seeing a geometric result with higher powers of $\pi$ (though I’ve been away from academic math for 15 years, so maybe my memory is going . . . .)  The paper definitely made me wonder if there was some relationship between these higher dimensional volumes and the sums that Euler solved.  Any relationship would be especially interesting since the volumes of the odd dimensional spheres have closed forms, too.     Some more reading on this subject is definitely in order!

Finally, if I’m writing about $\pi$, I have to mention another fun little closed form solution that I first saw in one of my courses in college.  It has always had a special place in my heart because it was such an incredible surprise to me that I made the decision right then and there that I would major in math:

Fun memories!