Circle π-PI (The Mysterious number)

The circle we know it from school time, but what is interesting about it? , the world you see is not linear always, a waterfall, ball throw, etc.. is in a curve, pi is always talking with a curve like structure, 4000 years ago the Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π.,

The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for π, which is a closer approximation.

Pi value was hidden in the universal nature we just digging it getting those numbers, A Supercomputer Calculated Pi to a Record-Breaking 62.8 Trillion Digits.-18-Aug-2021

Why PI is here?

To find the orbital speed of the satellite

v = (2 • pi • R)/T ,general equation for circular motion.

where R is the radius of the orbit.

Radius of orbit

The radius of orbit can be calculated using the above equation, which also involves the value of pi.

For Einstein’s field equation of general relativity:

It’s a theory of gravity. The basic idea is that instead of being an invisible force that attracts objects to one another, gravity is a curving or warping of space. The more massive an object, the more it warps the space around it.

Collisions give pi value

The collision of two box with two different weights , we get the collision of pi value why ? no one knows , why pi is here ? its just blew my mind paper

Buffon’s Needle

Buffon’s Needle is one of the oldest problems in the field of geometrical probability, It involves dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines on the page. The remarkable result is that the probability is directly related to the value of pi.

Online stimulation
https://mste.illinois.edu/activity/buffon/

Searching for the value of PI

circumference =2Π r ,diameter=2r .

For every small to big circle in this earth(I’m not sure about other earth in the multiverse) its circumference/diameter is equal to 3.14…, but why 3.141. ? why not other numbers?

Let’s think about it…

this letter Π on the left side is named from the Greek letter.

PI green word meaning circumference in English, π is pronounced as “pie” (/paɪ/ PY).

pi value never ends , like my friend’s ex GF Lament

Let’s go for a circle first…

we know Area=𝜋 r², consider that we do not know the pi value for now, so then how to find an area of a circle?

we can cut it down into equal pieces, and merging them together will form a rectangle we know the formula for the area of a rectangle is l*w.

the more slices, the more accurate area value we can get, therefore we cut that into n number of slices, and merging it will answer with some minimal error.

so the length of this rectangle is 1/2 circumference because they have cut down equally and merged, and then w is the radius, for the circumference also we need pi, so we need to get that value which will give us the area directly pi *r² so how to find the value of pi?

In general, we knew that any small to big circle its a circumference divided by diameter is pi.

we will make a circle and draw a square above that circle and inside that 6 sided hexagon. then considering that the area of that circle is within that two shapes.

So, by the above method, we came to know that pi is in between 3 and 4. but how to get a somewhat accurate value of it.

So by increasing the sides of the hexagon we can get a better pi value

now by increasing the hexagon into 96 sides we will get

𝜋 >3.1408 and 𝜋 < 3.1429

The first calculation of π was done by Archimedes of Syracuse (287–212 BC) ,The above 96 sides are done by Archimedes.
The last words attributed to Archimedes are "Do not disturb my circles"  (Latin, "Noli turbare circulos meos"; Katharevousa Greek, "μὴ μου τοὺς  κύκλους τάραττε"), a reference to the circles in the mathematical  drawing that he was supposedly studying when disturbed by the Roman  soldier.
Ludolph van Ceulen

Ludolph van Ceulen spent a major part of his life calculating the numerical value of the mathematical constant π, using essentially the same methods as those employed by Archimedes some seventeen hundred years earlier.

Ludolph van Ceulen discovered 35 digits in pi by calculating the Billions and Billions of side of hexagon

2⁶² =4quadrillion sides of hexagon===> for 35 point in pi

Then after many mathematicians tried many Billions side but is this really feasible way? , there he comes Isaac newton he discovers a better way for it

we know the binomial series

above it counts from 0 but what if? -1 or m/n form that was tried of Isaac.

if n==-1
then
(1+x)^-1 = 1 − 1x + 1x2 − 1x3 + 1x4 − 1x5 +..series continuous forever.
else if n==1/2
then
(1+x)^1/2 = 1+1/2∙x +(1/2)(1-(1/2))x^2/2!+....series continuous forever.
if n<1:series continuous forever.

for now solving above we will get an series like

Newton’s Series
(1-x²)¹/2= 1–1/2x² +1/8(x)⁴+1/16(x)⁶+….

for area finding using integration, you can check brilliant

To get a much better value of pi, newton tried the above integration not from 0 to 1 but 0 to 1/2, which will give make the calculation term faster.

after the integration(0,1/2) you will get pi =3.14161.., the more term you use in newton’s series, the more digits you will get

2⁶² =4quadrillion sides of hexagon===> for 35 point in pi (Ludolph van Ceulen almost spent is 25 years in his life)

equals to

newton’s series 50 term calculation ===> for 35 point in pi

there are many formulas that came after that Chudonovsky algorithm by Ramanujam, Machin-like formulae …

Pi value is always a mystery , it is not only talking with the circle , but all geometric space and time related things and many of them , for us the scale of one that we derived and we decided it, but maybe nature itself has scale of Pi as beginning one or starting point of something, it has opened many field in math, Quantum mechanics, etc and it will open many new as well...

ML Engineer