Great question! I think the closest thing to another physical application of 1^2 + 2^2 + ... + n^2 that I've come across is in finding the volume of a pyramid with a rectangular base without integration (kind of).

If you approximate the volume of the pyramid with the volume of a suitable step pyramid, then the above sum shows up naturally when you sum of the volumes of the n (cuboidal) steps. The limiting volume in n is then the volume of the original pyramid.

Sure, you could argue that it's just using the Riemann sum definition of Riemann integrals in a special case, but it's the sort of thing that I wouldn't be surprised if Archimedes did long before integrals were formalized.


Thanks! Yes, it's amazing when you start with a (more or less) mest messy formula or idea, and it ends up being really simple in the end. In some sense, I think it happens more often than it "should" when you deal with messy systems in physics, for example.

As a concrete example: Interactions between electric point charges (e.g. electrons) can be approximated in various ways with the Legendre polynomials. The n'th of these polynomials have a number of different series and contour integral representations that don't look particularly nice.

There is one very nice representation of the n'th such polynomial, though: Take the polynomial x^2 - 1, raise it to the n'th power, then take the n'th derivative of the result (and multiply it by a simple constant). Pretty easy to remember. It's also a bit odd in that we don't encounter derivatives of an order higher than 2 (or maybe 3) very often IRL.

"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram