posted by @ulaulaman about #PiDay #Archimedes #SrinivasaRamanujan and other mathematical curiosities
As you know, the $\pi$ is defined as the ratio of the circumference to its diameter. This number, which is transcendental, was, apparently, known since ancient times. There are, in fact, some Egyptologists who believe that $\pi$, or perhaps $\tau = 2 \pi$, was known to them since the age og the Giza's pyramid, built between 2589 and 2566 BC, because the relationship between the perimeter and the height is 6.2857.
There are no explicit proof of the fact that, at the time, Egyptian mathematics became aware of a number such as $\pi$, however, between 600 and 1000 years later on a Babylonian tablet it is geometrically established the first value of $\pi$: $25/8 = 3.1250$. From documents written more or less in the same period it can be deduced that also the Egyptians calculated the value of $\pi$, obtaining $(16/9)^2 \simeq 3.1605$.
Indian mathematics, however, seems a little late: in 600 BC on
Shulba Sutras, it is calculated for the $\pi$ value like $(9785/5568)^2 \simeq 3.088$, which will be updated later in 150 BC as $\sqrt{10} \simeq 3.1622$, which is a value much closer to the value calculated by the Egyptians.
A good approximation of $\pi$ value is in
Mishnat ha-Middot, a geometric treatise by
Rabbi Nehemiah: $3 + 1/7 \simeq 3.14286$.
The approximation, however, the most amazing not only for accuracy but also for the method is that proposed by
Archimedes, the italo-greek mathematician who invented the method of polygons in order to calculate $\pi$, a costant that for a millennium became known simply as the
Archimedes' constant.
He simply calculated the perimeter of polygons inscribed and circumscribed in a circle, thus obtaining a lower and an upper estimate of the value of the constant:
\[223/71 < \pi < 22/7\]
or
\[3.1408 < \pi < 3.1429\]
It's clear that his method of calculation is very modern and above suggests that Archimedes was well aware of the transcendental nature of the constant, which could be known only through approximations.
Today $\pi$ is known to
5 trillion digits and if you try to type the symbol $\pi$ on modern scientific calculators, the value they give you is, to the first decimal place, 3.14159265...
