[–] 10 points11 points  (5 children)

sorry, this has been archived and can no longer be voted on

If you take a 2D gaussian exp(-x2 - y2) the shape is both seperable in x and y (i.e. f(x,y) = f(x)*f(y)) and in r and theta. Because it is circularly symmetric the area (integral of the 2D function) is proportional to a circle of the same radius. Here the radius = 1 and the constant of proportionality is 1 (this also should take some thought). It is not surprising that a circularly symmetric function has an area proportional to pi. The sqrt(pi) comes from the fact that the function is also seperable in rectilinear coordinates (x,y), this means that the integral of the 2D function is just the product of the 1D functions. Since the 2D area is pi the 1D integral must be root pi.

[–][S] 2 points3 points  (1 child)

sorry, this has been archived and can no longer be voted on

I just reread your comment and I semi-retract my initial response. I guess the important thing about the random walk is that it gives a solution that is exponential; exponential functions are also the only ones I can think of right now that are separable in x,y and r,theta. I think I initially didn't catch the importance of this and. Very cool, thank you!

This is strange to me still, since it is a non-obvious symmetry-- one that is only really apparent when you consider the square of the integral. I hadn't thought much before about a system in N dimensions having (or not having) symmetry in additional dimensions. I am still curious if there is a physical reason why this symmetry "must" be in this case, but I think I appreciate your answer much more now! Thanks again!

[–][S] 1 point2 points  (1 child)

sorry, this has been archived and can no longer be voted on

Say I step outside my house, and once per second I take a step either to the North or to the South. After many hours, I record my final position. I do this every day, for many years, and make a histogram of my final positions.

I can use this data to find the probability, if I repeat this again, of ending back on my doorstep: (2*Pi)-(1/2)! It doesn't strike you as a little strange that Pi has come about in a walk in one dimension?

Forget the calculus: this was a result of purely physical measurements, addition and averaging. Why Pi!? Why this process!?

[–] -1 points0 points  (0 children)

sorry, this has been archived and can no longer be voted on

I don't know if it's physical, but a phase change of pi gives you a plus or minus 1, so initial position, phase change of 0 you go north, phase change of pi you go south.

[–]Plasma physics -2 points-1 points  (0 children)

sorry, this has been archived and can no longer be voted on

Essentially what he said, but with mathematical symbols: http://en.wikipedia.org/wiki/Gaussian_integral#By_polar_coordinates

[–][S] 7 points8 points  (0 children)

sorry, this has been archived and can no longer be voted on

A nice quote I found on the subject:

"THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

[–] 7 points8 points * (0 children)

sorry, this has been archived and can no longer be voted on

Well, the (unit covariance) gaussian distribution in any dimension is rotationally symmetric about it's mean. Pi pops up when you integrate over it because it pops up when you calculate the circumference of a circle, or in general the surface of a sphere in any dimension: You could first have integrated over the surface of spheres equidistant from the mean (over which the gaussian is constant), before integrating over this distance.

Gaussians with general covariance are just linear transforms of the unit covariance case, which adds a multiplicative factor to the integral.

The 1D case above arises because spheres on the real line are still spheres, even if they consist in just 2 points, I guess...

[–] 3 points4 points * (0 children)

sorry, this has been archived and can no longer be voted on

I don't think that there has to be a physical explanation of why you get pi. I.E. many integrations that are done using complex analysis, the pi's crop up because of Cauchy's Integral Formula, which does have a geometric interpretation.

[–] 1 point2 points  (0 children)

sorry, this has been archived and can no longer be voted on

Oh, what's a pi or two among friends?

[–] 4 points5 points  (0 children)

sorry, this has been archived and can no longer be voted on

But typically if a Pi shows up, there is something periodic or geometrical about the problem. I can't think of why it should show up in the Gaussian integral (the area of a normal distribution in 1-D).

Well, one way to solve the problem is to switch to polar coordinates.

Frankly, I long ago stopped viewing pi as something related to geometry. Instead, I view it as a Special Number, and geometry is just one of the many disciplines it shows up in. You might as well ask "Why does pi show up in geometry? What does geometry have to do with Gaussian integrals?"

[–]Particle physics 0 points1 point * (0 children)

sorry, this has been archived and can no longer be voted on

First pi doesn't only show up in circles and trig functions and the integral of Gaussians.

Take pi/4 = 1/1 - 1/3 + 1/5 - 1/7 + ... (-1)k/(2k+1) + ... (where k is an integer) or ei pi = -1. Both of these expressions on the surface have nothing to do with circles. I mean you are taking a summation of alternating numbers, or raising e to an imaginary exponent to get -1. Both of these can be motivated by geometric arguments: (arctan(1) = pi/4) and Taylor series of arctan x, and Taylor series of ex and sin x cos x, motivates ei pi .

See for multiple derivations of integral of a Gaussian e-x2.
http://www.york.ac.uk/depts/maths/histstat/normal_history.pdf

A particularly elegant proof starts with the Euler reflection identity ( link to proof ) Gamma[x] Gamma[1 - x] = pi/sin[pi x] with x = 1/2, we get Gamma[1/2] = sqrt(pi), and from definition of Gamma[x] = int(tx-1 e-t dt, t=0, t=inf) we see Gamma[1/2] = int( t-1/2 e-t dt, t=0, inf) and we note that if we let u = sqrt(t), then du = dt/(2sqrt(t)), so Gamma[1/2] = 2 int( exp(-u2) du, t=0, inf) so Gamma[1/2] = int(exp(-u2) du, u=-inf, inf).

Looking at the proof the Euler reflection identity you see it mirrors the previous proofs. You take an expansion of Gamma (actually 1/Gamma and the expansion is a product instead of sum of an infinite number of terms), and recognize that sin(x)/x has a similar expansion. Other than trig functions are involved, I don't see what you are looking for.

Also, can you think of any real geometric problem where you can come up with an answer of sqrt(pi) ? Other than the length of a square that has the area of a circle of radius 1 (a problem that is proven impossible by straightedge and compass ... e.g., outside the realm of ordinary geometry).

[–] 0 points1 point  (0 children)

sorry, this has been archived and can no longer be voted on

Edit: I'll restate the question to avoid more explanations of why the integral is Pi: Say you are performing some physical experiment, and some property of the measurement you make is Pi.

Lets say you didn't have a theory, but you are measuring a π. What you'd want is a theory that results in a π. The co-incidence of the measurement producing pi and there is math saying pi is a 'special number', means that it is unlikely that the constant you are measuring is a physical one.

On the other hand, the normal distribution just has to be normalized. We could say the same about e, however both e and π have accompanying constant, π the normalization factor, and e the uncertainty exp(1/(2 σ²)). One might think e and π are just our way of dealing with the numbers correctly.

Not sure if this answer is entirely satisfying, though..

[–] 0 points1 point * (0 children)

sorry, this has been archived and can no longer be voted on

Someone probably already said this, but I didn't see it ... maybe it's really stupid ... I don't know to be honest.

I am familiar with the multi-variable method of finding the answer, and I do not claim to know the answer to your question, but I have a guess.

From Euler's formula, we know that eix = cos(x) + isin(x), so if we let x = (ix2) then I believe that this becomes cosh(x2) - sinh(x2). I'm feeling too lazy to do this integral, but womframalpha says that it is sqrt(pi)*erf(t) (if the integral is from -t -> t). Erf(inf) = 1, so it agrees with the multivariable method. Perhaps this is the link to "periodic" functions you were asking about.

EDIT: The problem with this is of course that erf is defined with e-x2 ... so ...

[–] 0 points1 point  (0 children)

sorry, this has been archived and can no longer be voted on

My best guess is that it is like the strange connection between the exponential function and trig a la Euler's Formula. Trig is intimately connected with circles and pi, so it might not be too surprising that pi shows up in an integral of the exponential function, or at least not more surprising than Euler's Formula in the first place.

[–] 0 points1 point  (0 children)

sorry, this has been archived and can no longer be voted on

I wondered this as well - I remember when I was in college taking engineering courses and we see pi all the time there. And I still never quite understood how pi can have anything to do with inductance (or with distributions, same as you).

Even the ei*pi = -1 equation. I know this is true because I know how the to apply the imaginary coordinate system, but I don't get it at a fundamental level.

[–] 0 points1 point  (6 children)

sorry, this has been archived and can no longer be voted on

Because the exponential and periodic motion are linked by eix=isinx + cosx

[–] 0 points1 point  (5 children)

sorry, this has been archived and can no longer be voted on

Which is related to the identity ei*pi=-1, is it not? I don't think mathematicians understand this in a periodic or geometric context, it's something of a mystery. So the answer to the OP is " I don't know."

[–] 0 points1 point  (1 child)

sorry, this has been archived and can no longer be voted on

ok well then you surely know that that allows us to define sin x = (exi - e-xi) / 2

if you take the i out of this expression, you get hyperbolic sinh x = (ex - e-x) / 2

plotting these hyperbolic expressions onto a graph looks a lot like a gaussian dont it?

[–] 0 points1 point  (0 children)

sorry, this has been archived and can no longer be voted on

Okay, I'll buy that...so what's your point? Did I say something incorrect?

[–] 0 points1 point  (2 children)

sorry, this has been archived and can no longer be voted on

I don't think mathematicians understand this in a periodic or geometric context, it's something of a mystery.

It's not a mystery at all. It is initially surprising, and it requires a lot of study to see all the implications, is all.

I understand that it's not just your own personal notion; some mathematicians are fond of using excessive hyperbole and claim that it's a mystery, but that's just their mildly disingenuous way of saying that it's amazing and that the audience should look into it further.

Too many people believe, not just that it's a mystery, but also that that means that it's a bizarre curiosity, which badly misunderstands things, for actually it is one of the most important equations in mathematics, with countless vast and deep implications.

(I have here a complex analysis textbook that says that it is the most important, which may be true, but is a harder-to-defend position.)

[–] 1 point2 points  (1 child)

sorry, this has been archived and can no longer be voted on

Wow, thanks for that! My math teacher told me that :( this inspires me to go do some research!! I thank you, for you are a gentleman and a scholar.

[–] -1 points0 points  (0 children)

sorry, this has been archived and can no longer be voted on

My, what an unexpectedly positive response -- probably the most enthusiastic such I've ever gotten on Reddit. Thanks for the kind words.

I'm also gratified that you're inspired to research this; it is an utterly cool topic, and one of the most mind-blowing things to have been discovered by humankind.

Once you get far enough along to start feeling that this equation is natural, and indeed just could not be any different than it is, you'll have significantly increased your mathematical intuition and conceptual understanding.

Feel free to toss out questions to /r/math as you progress in your journey.

BTW there are many approaches to it, but FWIW it typically starts with the infinite series definition of ez: exp(z) = sum(k=0 to infinity) zk/k!

[–] -1 points0 points  (4 children)

sorry, this has been archived and can no longer be voted on

Pi shows up in the area of a circle. Integrals compute area under curves. Circles are curved. Therefore....

[–][S] 0 points1 point  (3 children)

sorry, this has been archived and can no longer be voted on

But x2 is certainly a curve; integral of x2 has no Pi...

I think this was a good idea though!

[–] 4 points5 points  (2 children)

sorry, this has been archived and can no longer be voted on

Because you don't get a circle from x2, but you do from exp(x2). When you multiply exp(x2)*exp(y2), you get exp(x2+y2) and your contours are circles.

If you multiply x2 and y2, you just get (xy)2.

[–][S] 1 point2 points  (1 child)

sorry, this has been archived and can no longer be voted on

If you rotated a parabola around the origin you would certainly get circles, one at each z coordinate. I only meant that in general, things can be curved, and you can integrate them, without getting a Pi.

[–] 1 point2 points  (0 children)

sorry, this has been archived and can no longer be voted on

It helps if your limits are circular. Try integrating x2+y2 in 2-D using polar coordinates over a circle and you'll get your pi.

The thing with the Gaussian integral is that since it goes to infinity, the limits can be made circular if desired.

[–] -3 points-2 points  (5 children)

sorry, this has been archived and can no longer be voted on

Set up the integral, I, as follows. Define I=integral(e-a*x2)dx = integral(e-a*y2)dy (True because x and y are just variables at this point). Examine the integral I2. I2=integral(integral(e-a(x2+y2)) dx)dy) Okay this is a hard one to do. Convert to polar coordinates (I see an x2 + y2!) I2 becomes integral(integral(e-a*r2 r dr d(theta))) where theta goes 0 to 2 pi, r goes 0 to infinity.

Evaluate theta integral, a 2pi pops out in front, other integral is (-e-(a*t) )/2a evaluated at infinity minus evaluated at 0. Multiply the answer by 2pi and you get pi/a.

With me so far? Here's the trick. You found I2. So I is just sqrt(pi/a). You're done. Thanks, ME400.

[–][S] 1 point2 points  (4 children)

sorry, this has been archived and can no longer be voted on

I understand how to do the integral. Did you read the original post?

[–] 0 points1 point * (3 children)

sorry, this has been archived and can no longer be voted on

Aye, x and y are just dummy variables. You're going to polar coordinates because you're using the distance r "x2 + y2!", distances in polar coordinates pop out a pi because you're looking ALL AROUND YOU (2pi)

Edit not really dummy variables, but you know what I mean. You get the pi cause you're looking everywhere around you. At least I'm pretty sure that's why if my understanding is correct. (NOT A MATH MAJOR--ENGINEER UNDERGRAD)

[–][S] 1 point2 points  (2 children)

sorry, this has been archived and can no longer be voted on

You get the pi cause you're looking everywhere around you.

GAH! I understand how to do integrals, and integrals in polar coordinates, and even this exact integral in question. You seem to be missing the entire point of my question.

And for the record, "cause you're looking everywhere around you" would be how to explain a factor of 2*Pi to a two-year-old. Not totally helpful for people who have already taken and understand calculus.

[–] 6 points7 points  (1 child)

sorry, this has been archived and can no longer be voted on

Well how else would you characterize the fact that you're looking everywhere around you without using pi? What kind of more indepth physical meaning are you searching for; isn't this precisely the kind of explanation that you want? You've taken and understand calculus. You know that integrals come about because of looking at small differences. I ask you--differences in WHAT, in this case? Well you have differences in space extending to infinity in all directions, and then you have differences in angle extending the range 0 to 2*pi. You could characterize it in terms of degrees, but then you're just dancing around the point. The fact of the matter is, no matter how you look at it, since there's a radial symmetry when coming up with the square of the gaussian integral you're asking about. There's no known elementary functions that intuitively give you pi for the gaussian integral, that's why you have to look at the distribution of the gaussian over x AND y, then use symmetries in the problem to get the original integral.

In short; the square if the integral you're asking about has radial symmetry and you would do it using polar coordinates, so you get pi. You claim you already knew that, so what are you asking about? "Why does pi show up in a 1 dimensional random walk?" --Because pi shows up in two dimensional random walks, and due to symmetry the 1 dimensional random walk is the square root of the two dimensional random walk.

You're asking for a physical reason of why an integral has a pi in it, then belittle an attempt to explain it using integrals and integration techniques because you understand calculus beyond my "two-year-old" explanation. What kind of explanation would satisfy you?

[–][S] 2 points3 points  (0 children)

sorry, this has been archived and can no longer be voted on

OK, I am sorry because I think I was a bit rude. I apologize. But it wasn't really clear in your first two comments that you were trying to explain anything other than just how to do the integral, and I misinterpreted the way in which you were trying to explain it as possibly condescending.

Because pi shows up in two dimensional random walks, and due to symmetry the 1 dimensional random walk is the square root of the two dimensional random walk.

This part here, though, I believe is one of the most correct and concise statements on this thread. Thank you, this was very well put.

[–] -1 points0 points  (0 children)

sorry, this has been archived and can no longer be voted on

[–] -3 points-2 points  (12 children)

sorry, this has been archived and can no longer be voted on

If I remember correctly, it's a result of the change-of-variables you need in order to do the integral. That is, again if I'm remembering this right, square the integral, change one of the variables to 'y' so you're integrating over S S exp[-x2]*exp[-y2] dx dy. Then change to polar coord's (r, theta). From there it falls out.

[–][S] 2 points3 points  (11 children)

sorry, this has been archived and can no longer be voted on

Yes, you are correct. I understand how the integral comes about. I'm wondering if there is a physical reason, though, not a mathematical one.

[–] 2 points3 points  (1 child)

sorry, this has been archived and can no longer be voted on

I think you're looking for a geometrical reason, not a physical one. Geometry is part of math, and can exist outside of physics. Btw, you should post your question on the math reddit, you might get better answers. I must say I'm interested in the answer, too.

[–][S] 1 point2 points  (0 children)

sorry, this has been archived and can no longer be voted on

Geometrical would be OK. Then I can at least try to come up with some reason why a random walk, etc. is analogous to the geometrical situation. Since you can have a random walk in 1-D, though, and the Pi will still show up-- I'm having a hard time making a geometric analogy.

[–] 1 point2 points * (0 children)

sorry, this has been archived and can no longer be voted on

I think the key here is translating the mathematical steps we took to do the integral into the physical situation of a random walk. With the math, we couldn't do the integral in 1D, we had to go to 2D, so we have to do that in the physical random walk too. So we randomly walk north-south, and then west-east and ask what's the probability that we ended up at the starting point after both random walks? That's the probability of ending up at the same latitude times the probability of ending up at the same longitude, which is the square of the probability for the 1D case. So we can convert the 1D random walk to the 2D without talking about calculus, just about geometry and probability.

So after that, what did we do in the math? We converted to polar coordinates. So back to the physical situation: walking north-south and then east-west is the same as just walking in any direction.

And how did the factor of pi come in? We integrated over theta from 0 to 2pi. I think what this means in the physical random walk is that it doesn't matter what angle you're at when you get back to the origin, i.e., the origin is like a really tiny circle, and it doesn't matter where on the circle you are.

Edit: that last part isn't right. But you can see there are angles/circles involved somewhere when you think about the 2D random walk in polar coords.

[–] 0 points1 point  (7 children)

sorry, this has been archived and can no longer be voted on

Because the natural world follows the rules of Mathematics.

[–][S] 2 points3 points  (6 children)

sorry, this has been archived and can no longer be voted on

But... Pi doesn't show up in all integrals, why this one?

[–] 0 points1 point  (5 children)

sorry, this has been archived and can no longer be voted on

Because there's no theta dependence when you switch to polar coord's. I'm not aware of any physical meaning 'why' it's there. It's just a number. A useful one, but a number all the same.

[–] 1 point2 points  (4 children)

sorry, this has been archived and can no longer be voted on

It's not just a number. It's a very specific number: the ratio of the circumference to the diameter of any circle! Like lisatomic, it also creeps me out to see it eerily show up in things which have, at face value, absolutely nothing to do with circles.

[–] 1 point2 points  (3 children)

sorry, this has been archived and can no longer be voted on

But it may have something to do with being periodic (or at least represented by a periodic function) and Pi is all over that like white on rice in a snow storm.

[–][S] 0 points1 point  (2 children)

sorry, this has been archived and can no longer be voted on

Yes, this sort of thing is what I'm getting at-- what is periodic about a normal distribution?

[–] 1 point2 points * (1 child)

sorry, this has been archived and can no longer be voted on

Yes, this sort of thing is what I'm getting at-- what is periodic about a normal distribution?

You want to know what's periodic about a normal distribution. Now, how do you get the periodic components of something? You Fourier transform it.

So what happens when you Fourier transform a normal distribution, then?

You get... Er, a Normal distribution! The Gaussian integral in particular is an eigenfunction of the Fourier transform, with eigenvalue 1...

Food for thought?

[–][S] 0 points1 point  (0 children)

sorry, this has been archived and can no longer be voted on

I've thought of this :)

I can not continue though! Thank you, I think it may help.