Mikael Passare’s enthralling visual proof of the Basel problem. For those who have not heard, this is the problem on which our Master, Leonhard Euler, famously cut his teeth.

Although Passare’s method is advanced, the main idea relies mostly on trigonometry. Getting from the geometric insights to the calculations is pretty sophisticated. For our purposes, I hope a bit of informal synopses will suffice:

First, Passare describes coordinate systems (radial and angular bipolar coordinates), which are given by a clever construction of a triangle. Solving the triangle using the sine and cosine rules, Passare defines a bijective map and inverse map to and from these coordinates.

Passare then applies a logarithmic change of scale. This generates an amoeba (in complex analysis parlance). The amoeba is called U.

The first quadrant of the amoeba is U_0. Its upper boundary is the dotted line in the image above (the exponential function in terms of (x,y)). Solving for y and integrating from zero to infinity is one way to find the desired sum. This appears in the equality below.

Another way is double integrating over the region U_0, which likewise features in the equality below. This shockingly simple integral (found using logarithmic radial bipolar coordinates), as well as the one right of it (log-angular bipolar, bounded by T_0), are the central results of the paper.

The leftmost sum in the bottom equality is represented above as piled boxes– squares of area 1/(n^2). Finding their total area is the Basel problem.

Passare does some incredible acrobatics to show that the equally-shaded regions in the above figure have equal area (by proving that there is an area-preserving map between them). The sum of the improper integral, below, ultimately describes this phenomenon.

The promised equality:

You should read the paper. A remarkable insight is seeing T, which is a right triangle with base pi. Hence its area is (pi^2)/2. It appears in such a way that we need only 1/3 of it (sketchy, I know, read the paper :D). This visually and immediately affirms the famous Basel sum, (pi^2)/6.

Read the paper. <3 Really, it’s only 8 pages! Now, I don’t *understand* it–but the more and more I read it,* I get it*. Stretch your mind! Get the .pdf here.

Mathematics is beautiful. <3