Pi stories: the Cyclometricus and other tales

Fifth period

It was 1621 when the Cyclometricus by Willebrord Snellius, a pupil of Ludolph van Ceulen, was published. Snellius proved that the perimeter of the inscribed polygon converges to the circumference twice than the circumscribed polygon. As a good pupil of van Ceulen, Snellius managed to get 7 decimal places for the $\pi$ by using a 96-sided polygon. His best result, however, was 35 decimal places, which improved his master's result, 32.
The next improvement is dated 1630 by Christoph Grienberger, the last mathematician to evaluate $\pi$ using the polygon method, while the first successful method change came out thanks to the british mathematician and astronomer Abraham Sharp who determined 72 decimal places of $\pi$, of which 71 correct, using a series of arctangents. A few years later, John Machin improved Sharp's result with the following formula and that allowed him to achieve the remarkable result of 100 decimal places! \[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}\] Machin's approach proved successful, so much so that the slovenian baron Jurij Vega improved on two occasions the above formula obtaining a greater number of decimal digits of $\pi$, the first time in 1789 with a formula similar to Euler's one \[\frac{\pi}{4} = 5 \arctan \frac{1}{7} + 2 \arctan \frac{3}{79}\] then in 1794 with a Hutton-like formula \[\frac{\pi}{4} = 2 \arctan \frac{1}{3} + \arctan \frac{1}{7}\] The arctangent era continued with William Rutherford \[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{70} + \arctan \frac{1}{99}\] and with Zacharias Dase \[\frac{\pi}{4} = \arctan \frac{1}{2} + \arctan \frac{1}{5} + \arctan \frac{1}{8}\] Finally comes the british William Shanks that pushing the full potential of the Machin's formula managed to get 707 decimal places, of which only 527 were correct after Ferguson's controls in 1946. Here, however, we are going in the era of mechanical calculation, prologue to computer era.

Vibrating string

Frazz by Jef Mallett

Many of $\pi$'s "appearances" in mathematical and scientific formulas are closely linked to geometry. However, there are some applications where this is not correct. For example in the case of a vibrating string within the unit range $[0,1]$.
The modes of vibration of a string are the solutions of the following differential equation \[f'' (x) + \lambda f(x) = 0\] where $\lambda$ is a strictly positive number associated with the eigenvalue. Then, said $\nu$ the wave number, $\lambda = \nu^2$.
It is observed that $f(x) = \sin (\nu x)$ satisfies the boundary conditions and it is the result of the differential equation for a vibrating string when $\nu = \pi$, which thus coincides with the wave number of the fundamental mode of the vibrating string.

Pi in the sky

One of the main features of $\pi$ is its invasiveness in nature. It will be because it is defined as the relationship between the circumference and its diameter, and therefore the result of spherical symmetries, or it will be for its intrinsic charm, perhaps linked to the infinity of its decimal figures. In every case we also find it in the sky, where it loses the property of number but acquires that of... stellar name!
There are in fact in the firmament some stars that are identified with $\pi$: for example the six stars that make the Orion' shield or the double star Pi Bootis which is located just below Arturo.

The Orion Constellation realized with Stellarium - via Universe Today

We can also find $\pi$ in one of the most important cosmological equation, the Einstein's general relativity equation: \[R_{ab} - \frac{1}{2} R g_{ab} + \Lambda g_{ab} = 8 \pi G T_{ab}\] It is on the right side, together with the universal gravitation constant, in the term that identifies the energy of matter contained in the universe. The term on the left, however, is linked to the curvature of spacetime, or to its geometry, which perhaps makes the presence of $\pi$ on the right side of this equation a bit strange.
It is known, however, that mathematics is full of mysterious mysteries, and the fun part is just revealing them!