Estimating Pi with the Digits of Pi // #PiDay2023

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I say "modified" because of course the whole point of an actual Monte Carlo simulation is to choose the points at random, and then check if they lie within a particular area. In this case, the area is a circle inscribed inside of a square. The ratio of the area of the circle to the area of the square is π/4, so by multiplying the proportion of red points to all points by 4, we should get a decent estimate of pi.

Happy Pi Day 2023!

#piday2023 #piday #montecarlo

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[puniopenetrante]

Engineers: pi=e=3 take it or leave it.

[IGChannel]

How this works:

Let red circle radius = r
=> square side = 2r
square area = 4r^2
red circle area = pi r^2

4 x pi r^2/4r^2 = pi

The circles area is represented as the red digits
The square area is represented as all digits

As it uses more digits the approximation will become better

[바보똥멍충이]

It's about geometric probability. If you make a dot in square randomly, the probability that you made dot in a circle which inscribed in square is area of inscribed circle divided by whole area of square. If the radius of square is r, that probability is pi*r^2/4r^2=pi/4.
And this means, since the probability is equals to ratio between red dots and all dots, so, that ratio*4=pi.
It's interesting, thx for nice video.

[BrianSpurrier]

In a way you could say that 4 is the “pi for squares” in the sense that if you take an inradius (aka apothem) of the square, it’s area and perimeter can be calculated as
A = 4 * r^2
P = 2 * 4 * r
Which match the equations for a circle, with 4 in place of π.
For any regular polygon this value is
Π(n) = n*tan(π/n)
A = Π(n) * r^2
P = 2 * Π(n) * r

[DrDeuteron]

a Monte Carlo simulation uses randomly placed digits, vs gridded digits. No big deal for 2 dimensions, but if we were looking at a 99-D hyper surface in a100 dimensional hypersphere, the Monte Carlo converges as N^(1/2), while the grid converges as N^(1/200)...A LOT SLOWER.

[CapAnson]

I made a computer program to calculate it this way once.. it starts taking exponentially more points to get each additional digit of pi and its a pretty slow way to calculate it.

[arvindsrinivasan]

Just wanted to say that this is not like a Monte Carlo simulation because the placement of the digits is a grid. Either way it will converge to pi, but without the randomness it will happen a lot faster.

[WildOtter]

How to estimate pi :
- First we need the first 1000 decimals of pi
- Then we put these numbers in a square, and color some in red to form a circle
- We calculate the ratio of red numbers / blue numbers and multiply by 4
Congratulations, you now estimated the first 10 decimals of pi.

[승수노-ze]

pi=4×circle area/square area

square area 1= circle area x?
2=x?
3=x?
4=x?
5=x?
...
square area= 1 2 3 4 5 6 7 8 9 ...,
square area=1,
circle area x=pi×1/4
square area=2,
circle area x=pi×2/4
square area=3,
circle area x=pi×3/4
...

[miked]

You can use 22/7 for normal stuff. It is good appoximation.

[Lialvas]

dont let this distract you from the fact that π^2 is almost equal to the g (the acceleration due to gravity of earth, i was mind blown when my teacher first asked me to substitute π^2 for g)

[jakobr_]

It would be cooler if you *added* the red digits and divided by the expected value (4.5) times the total number of digits. Then you’d *actually* be using pi. Only issue is that we don’t actually know if pi is “normal”.

[theawakenedphoenix]

what if you fill the whole square with dots

[pcrombie]

try using 22/7ths you might find a better answer

[billkillernic]

this video would be far more interesting if you also have added the numbers inside the circle and divided that with sum of the blue numbers

[danser_theplayer]

So the reason pi is so goddamn long is because the math people decided to start counting it from a *square* untill it gets rounder and rounder? But that would require infinite amount of points added to the 4 base points of a square to have perfectly round thing.
Somebody do it different pls.

[guzmat]

Cook, I did something very similar in my channel ... cheers

[sumdumbmick]

pi is the 4 of circles. so yeah, pi kind of is 4.

check it:
- pi/4 is the ratio of the perimeter of a circle to the perimeter of the square that circumscribes it, because the square has 4 sides
- pi/4 is the ratio of the area of a circle to the area of the square that circumscribes it, because pi*r^2 cuts the circle into 4 pieces. note that if you found the area of the square the same way you'd end up with an infinite regress that yields a convergent infinite product of the form 4*4*4*4*4*4*4*4... which to my eye appears to be a complete fucking paradox
- pi/4 is the angle about which the trig functions are reflected, so cosine = sine reflected about pi/4, cotangent = tangent reflected about pi/4, cosecant = secant reflected about pi/4, and this fact allows for really nice Taylor series for computing pi that converge really fast, except they actually compute pi/4 and people just multiply the result by 4 at the end, because indoctrination be like that, and notably you're having to do the same thing here

[techie]

But don’t you have to know pi in order to draw the circle in the computer simulation in the first place?

[yiutungwong]

π = 2 in Riemann Paradox And Sphere Geometry Mathematical Systems Incorporated ❤😂🎉😢😮😅😊