Sum of squares

There is a famous anecdote about the precocious mathematician Carl Friedrich Gauss, who at the age of eight was tasked by his teacher J.G. Büttner to calculate the sum of the first n positive integers; that is, the sum S = 1 + 2 + 3 + ... + 100.

To the astonishment of his teacher and his assistant, Gauss provided the correct answer of 5,050 within seconds. Gauss' "trick" was presumably to realize that the terms in this sum could be written in reverse and recombined with the original sum in a nice way. Specifically, we have

S =  1   +  2  +  3  + ... + 100

and

S = 100 + 99 + 98 + ... +  1.

By looking at the terms above that line up vertically, we can see that

1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, ..., 100 + 1 = 101.

That is, each of the 100 pairs of terms that line up has a sum of 101. Thus, by adding these two sums together, we have that

2S = 101·100, or S = 5050.

Recently, I came across a very nice generalization of this idea for the sum of the first n squares; that is, 1² + 2² + 3² + ... + n². Apparently it's a quite obscure result that isn't really (IMO, anyway) suitable for being presented in a textbook, so I created an animated version of it. Enjoy:



(Do you recognize at any of the equations in the introduction?)

Interesting. Besides stacking cannonballs (and wouldn’t they stack better in triangle bases?), have you encountered many real world uses for the sum of the first n squares?

Interesting. Besides stacking cannonballs (and wouldn’t they stack better in triangle bases?), have you encountered many real world uses for the sum of the first n squares?

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.

That was super interesting I really enjoyed the way you presented that. I love when sums can collapse down to a simple equation, it really is beautiful to watch.

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.

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.

I feel this is a side effect of us having very beautiful mathematical language at our disposal. Our understanding of mathematics has come so far that even the most complicated aspects can be expressed in a simplistic form that is easy to understand. It's one of my favorite things about math.

That was super interesting I really enjoyed the way you presented that. I love when sums can collapse down to a simple equation, it really is beautiful to watch.

your animations / voice / format is excellent, but the math is too advanced for dumbasses like me. I suggest your new video: "Top 10 Titties According to the Golden Ratio"