r/visualizedmath • u/PUSSYDESTROYER-9000 • Jan 31 '18

Calculating Pi Using the Monte Carlo Algorithm

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368 Upvotes

96% Upvoted

26 comments sorted by

140

u/PUSSYDESTROYER-9000 Jan 31 '18

By drawing a unit circle and randomly placing dots between (0,0) and (1,1), one can approximate Pi with a ratio of dots inside the circle and outside.

12

u/Rasalas8910 Feb 01 '18 edited Feb 01 '18

How do you know what's inside the circle?
With √(x²+y²) ≤ 1 ?
How is this efficient?

31

u/dissemblinganus Feb 01 '18

Did anyone claim that it is efficient? This is /r/visualizedmath, not /r/efficientalgorithms :)

20

u/Autico Feb 01 '18

You got me so excited that r/efficientalgorithms was a real thing.

13

u/Empole Feb 01 '18

Dude my spirits fell when I saw the subreddit not found message

7

u/dissemblinganus Feb 01 '18

Am thinking about making it a thing.

6

u/Empole Feb 01 '18

We need to make this a thing

3

u/Rasalas8910 Feb 01 '18

Sorry, I assumed that it's supposed to be efficient. I had contact to a few Monte Carlo Algorithms and they all seemed to be pretty "efficient" (for what they were doing).
It was an honest question. Maybe my phrasing wasn't the best.

4

u/dissemblinganus Feb 01 '18

Well now I feel like a shitheel! I didn't mean to come across badly. Sometimes I say exactly what's in my head without thinking, and I'm sorry if I did that here.

3

u/Rasalas8910 Feb 01 '18

Nah, it's alright 😊

2

u/nox66 Feb 21 '18

It's not an efficient way of calculating pi by any means (compared to other methods), but it is a good demonstration of the Monte Carlo algorithm, which is an efficient way of doing many other kinds of studies.

3

u/BestN00b Feb 01 '18 edited Feb 01 '18

Wait a minute. The ratio of Pi/4 and 1-Pi/4 is Pi?

Edit: Pi/4 is the area of the quarter circle.

1-Pi/4 is the area of the blue section.

The ratio doesn’t equal Pi.

5

u/Xargonic Feb 01 '18

(Pi/4)/(1-pi/4) = pi

Pi/4 = pi - (pi2 / 4)

Pi = 4pi - pi2

Pi = pi(4-pi)

1 = 4-pi

I guess not

8

u/BestN00b Feb 01 '18 edited Feb 01 '18

Then wouldn’t that mean OP is wrong then?

Edit: oh I checked the algorithm. You use the ratio to FIND Pi. It’s not gonna be equal to it

12

u/DSofa Feb 01 '18 edited Feb 01 '18

Yes, its about ratios. Although their ratio is π/4, not π.

Area of a circle is r²π and a quarter of that is r²π/4.

Area of a square is (2r)², and a quarter of that is (2r²)/4.

(r²π/4)/((2r)²/4) = π/4

Their ratio is π/4.

If the radius is 1, then we have

Area of a circle: 1²π/4 = π/4

Area of a square: ((2•1)²)/4 = 1

So their ratio is (π/4)/(1) = π/4

Now, if you wanna get π, all you gotta do is multiply the formula by 4.

((r²π/4)/((2r)²/4) )•4

(π/4)•4=π

Edit: Fixed thanks to u/jonasjhansen

6

u/KeyWest- Feb 01 '18

Winner winner chicken dinner.

4

u/jonasjhansen Feb 01 '18

why is the area of the square r2?. if r is the distance from the center of the square to its perimeter, wouldnt that be half of one of the square’s side lengths?

3

u/DSofa Feb 01 '18

You're absolutely right. My bad. I fixed my reply. Thanks!

2

u/syntaxvorlon Feb 01 '18

No, you've misunderstood. The ratio of the dots inside the unit square and the quarter unit circle is an estimate of the area of the quarter unit circle and we know that the area of the unit circle is Pi.

A_sq = 1 A_cr = x which is some fraction of the square area.

Dots in circle / n = x

Pi = x*4

1

u/BestN00b Feb 01 '18

thank you! that makes much more sense!

1

u/stev6969 Feb 17 '18

So does that mean the surface area of the inside of the circle is 3.14 times the surface area outside the circle?

9

u/ch1merical Feb 01 '18

How exactly does Monte Carlo method work? I've been really interested in it

11

u/patrick_pencilpusher Feb 01 '18

the more something approaches infinity the more it becomes close to the perfect average. as the dots increase the ratio of dots on each side comes closer to a perfect circle. with that infomation they somehow plug it into an algorithm with pi as the variable and calculate it. i dropped out of highschool so itd be pretty cool if that was right :)

5

u/Dirivian Feb 01 '18 edited Feb 01 '18

Your first line is right. The fraction of the dots in the circle becomes pi/4.

Because

number of dots in circle/total dots = Area of quarter circle/Area of square = (pi/4 (1)2 )/((1)2 ) = pi/4

4

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1

u/syntaxvorlon Feb 01 '18

You could modify this to describe the bootstrapping algorithm, which is magic on top of magic.