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