The Strange Occurrence of Pi Everywhere

[Mindscrape]

This has probably already been brought up at some point, but does anyone else think it is strange how much pi, the ratio of circumference to diameter, occurs in so much that has nothing to do with circles? I mean, why should an electric potential between two grounds have any significance with respect to the ratio of circumference to diameter (through the Fourier series)? Even more so, suddenly e, an equally strange appearance, comes into into play. They appear all over the place too, oscillations, orbits, etc. Mankind made a peculiar invention with math.

You more mathy folks might think its totally normal within the math context, but no one is going to look at the physical situation of two grounded conductors (without already seeing the solution) and say, "Oh, I bet that 2.72 and 3.14 play a big part."

 

[neutrino]

Because we tend to approximate a lot of things to perfect circles and spheres. Even though some things may not actually involve circles or spheres, if you trace the steps backwards, you'll come across them somewhere.

As for e, it's usually the base of the logarithm used in calculus.

 

[ice109]

This has probably already been brought up at some point, but does anyone else think it is strange how much pi, the ratio of circumference to diameter, occurs in so much that has nothing to do with circles? I mean, why should an electric potential between two grounds have any significance with respect to the ratio of circumference to diameter (through the Fourier series)? Even more so, suddenly e, an equally strange appearance, comes into into play. They appear all over the place too, oscillations, orbits, etc. Mankind made a peculiar invention with math.

You more mathy folks might think its totally normal within the math context, but no one is going to look at the physical situation of two grounded conductors (without already seeing the solution) and say, "Oh, I bet that 2.72 and 3.14 play a big part."

periodic motion can be described by a circular coordinate system. do i really need to explain orbits? i don't know about Fourier series and potential between 2 grounds but field drops off from a charge as 1/r^2 which sets up circlular equipotentials. you want a really weird occurrence of pi? look up some of the series that sum to pi. e.g.

$$\pi = \frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots$$

 

[AlephZero]

But if anybody can give a phyisical "explanation" as to why $e^{i\pi}+1=0$, that would be interesting.

Sure, pi relates to Euclidean geometry, and e relates to physical growth and decay phenomena - in a sense, it defines a natural measuremt of time to describe the processes.

But why should pi and e relate to each other through a simple equation? I guess the association between the complex plane and 2-D Euclidean space must figure somewhere in the link...

 

[mjsd]

This has probably already been brought up at some point, but does anyone else think it is strange how much pi, the ratio of circumference to diameter, occurs in so much that has nothing to do with circles? I mean, why should an electric potential between two grounds have any significance with respect to the ratio of circumference to diameter (through the Fourier series)? Even more so, suddenly e, an equally strange appearance, comes into into play. They appear all over the place too, oscillations, orbits, etc. Mankind made a peculiar invention with math.

You more mathy folks might think its totally normal within the math context, but no one is going to look at the physical situation of two grounded conductors (without already seeing the solution) and say, "Oh, I bet that 2.72 and 3.14 play a big part."

Perhaps, the sphere/spherical shape is everywhere, and that's why $$\pi$$ pops up so often. Not a coincident either (if you believe in that fact that our world is full of symmetries and symmetry is indeed the guiding principle in our axioms/laws and logics). Think of a perfectly symmetrical surface in 3D you get the sphere. in fact in all dimensions, you get some kind of a hypersphere and your old friend pi shall certainly drop by for a visit. How can anything be evenly distributed around a point? you get the sphere again. How to get reflection symmetry in all direction?... the sphere; rotation? sphere... the list goes on. It is this symmetry principle in stuffs, the space they reside in and their interactions that makes the "sphere"/circle and hence $$\pi$$ important.

 

[HallsofIvy]

But if anybody can give a phyisical "explanation" as to why $e^{i\pi}+1=0$, that would be interesting.

Sure, pi relates to Euclidean geometry, and e relates to physical growth and decay phenomena - in a sense, it defines a natural measuremt of time to describe the processes.

But why should pi and e relate to each other through a simple equation? I guess the association between the complex plane and 2-D Euclidean space must figure somewhere in the link...

There is no reason to expect or want a "physical" explanation of a purely mathematical expression. I suspect that any "physical" study of $\pi$ would founder on approximation issues and "physically" there is nothing special about e. Any exponential can be written using any base.

 

[AlephZero]

What I was thinking was: pi and e are related to elementary Euclidean geometry, of circles and rectangular hyperbolas for example. The complex plane can be represented in Euclidean geometry. So, is there a geometrical (non-calculus-based) proof in terms that Euclid, Archimedes, etc would have recognized, that $e^{i\pi}+1 = 0$?

Obviously this is equivalent to a purely geometrical demonstration that $e^{i\theta} = \cos\theta + i\sin\theta$.

The Greeks were quite happy with (semi-pictorial) geometrical arguments about limits of areas, etc...

Apologies for bad use of the word "physical" - working as an ME, Euclidiean geometry is pretty much isomorphic to physical reality!

 

[HallsofIvy]

What I was thinking was: pi and e are related to elementary Euclidean geometry, of circles and rectangular hyperbolas for example. The complex plane can be represented in Euclidean geometry. So, is there a geometrical (non-calculus-based) proof in terms that Euclid, Archimedes, etc would have recognized, that $e^{i\pi}+1 = 0$?

Obviously this is equivalent to a purely geometrical demonstration that $e^{i\theta} = \cos\theta + i\sin\theta$.

The Greeks were quite happy with (semi-pictorial) geometrical arguments about limits of areas, etc...

Apologies for bad use of the word "physical" - working as an ME, Euclidiean geometry is pretty much isomorphic to physical reality!

Excellent! A purely geometrical demonstration would be mathematics and not at all "physical". I doubt that there would be such a demonstration that Euclid and Archimedes would have recognized because you will have to have "i" in there somewhere- and they wouldn't have recognized that.

(I suspect a physicist working in general relativity would NOT say "Euclidean geometry is pretty much isomorphic to physical reality"!

 

[Chris Hillman]

If You Like Simplicity, Which Mathematicians Do, You Can't Avoid Pi

This has probably already been brought up at some point, but does anyone else think it is strange how much pi, the ratio of circumference to diameter, occurs in so much that has nothing to do with circles?

This is really a FAQ, but it's a good question nonetheless.

"Why" questions can be problematic if you are not expecting the kind of explanation likely to be offered by mathematicians--- which brings us to another FAQ, "What is mathematics?". The standard answer is that the definining characteristic of mathematical discourse is the notion of proof, and I don't disagree, but let me toss another idea into the mix. I like to define mathematics as the art of reliable reasoning about simple situations without getting confused. According to this definition, proof is merely a means to the end of reliable reasoning while avoiding confusion.

If you accept this, it follows that mathematically simple situations will turn up more often in the most successful mathematical theories, which are the ones most likely to appear in undergraduate math curricula. And be it noted: one component of what we call "mathematical genius" is the ability to recognize a simple phenomenon masquerading as an apparently complicated phenomenon.

Given this, the explanations I prefer are information theoretic: we define appropriate notions of complexity of differential equations and then show that the simplest differential equations are those which give rise to familiar trig, hyperbolic trig, exp, and (at the next stage) the best known special functions, such as Legendre, Bessel, elliptic, and hypergeometric functions. But of course we should not be too surprised if $\pi$ turn ups when we study circular trig functions (for example when integrating over the circle). And we can expect circular trig functions to turn up more than hyperbolic trig functions because the circle is compact.

(A principle which is challenged by the importance of the Lorentz group in physics, but in the interests of simplicity, let's ignore that. And one could turn this reasoning around and adduce the fact that by our definitions the ODE $\ddot{x} + x = 0$ is particularly simple as evidence that our definitions of "complexity" are reasonable, which might cause some to doubt that we are not simply talking in circles--- no pun intended!)

We should recall here the principle that the simplest differential equations tend to model more than one situation. This is the well known "paucity of low dimensional models" phenomenon. Also, this style of reasoning is not limited to differential equations; similar remarks hold for algebraic plane curves.

I note too that probability concerns measures, and when we normalize Lebesque measure on the unit circle, we introduce a factor of $1/\pi$, so we shouldn't be terribly surprised that this constant is involved in the computation of various probability problems in the real plane. Indeed, the most efficient way to derive the most often encountered probability distributions (Gaussian, Poisson, and so on) is via the Principle of Maximal Entropy. Many of these involve $\pi$ in some way, if you like because these distributions are expressed using the "usual suspects" such as the exponential function.

Trig functions also arise in higher dimensions because n-dimensional unit spheres contain (n-1)-dimensional unit spheres. Very often in mathematics we are interested in how functions decay with distance from "the origin", and analyzing such behavior leads naturally to harmonic decompositions in which integrations over spheres play a central role. (This is one way to explain the phenomenon mentioned by Halls, that spheres are everywhere in mathematics.) So here too we should expect $\pi$ to play a role, and it does.

All in all, it is not very surprising that the same special functions, and therefore the same constants associated with special functions, arise all over the mathematical map. It all comes down to the unity of mathematics, and ultimately one way of explaining the happy meeting of algebra, analysis, and geometry is the information theoretic principle that leads us to expect a preference for simple models, of which there are not many, hence these models are familiar friends to undergraduate math students.

If you read German, you can look in the papers of Hurwitz for some famous examples of this kind of reasoning.

As for Euler's formula, this is less mysterious if you know about Cayley-Dickson algebras, where the starting point is factoring a simple partial differential equation. In the simplest nontrivial cases ("elliptic, parabolic, and hyperbolic"), we obtain respectively circular trig, parabolic trig, and hyperbolic trig together with the appropriate "adjoined unit" (not a real number) obeying $e^2 = -1, \, e^2 = 0, \, e^2 = 1$ respectively, and with isotropy groups $SO(2), \, R, \, SO(1,1)$. Only the first is compact, so again by the principle of analysis which says that compactness tends to be associated with simplicity, we should expect circular trig to be more important than parabolic or hyperbolic trig. And it is.

(The fact that parabolic trig is associated with Galilean relativity and hyperbolic trig with special relativity challenges this notion, but overall, compactness wins, in my estimation.)

 

[ice109]

chris hillman who are you!

 

[matt grime]

Someone about whom one can find details using google quite easily (Chris's contributions to Wikipedia and Usenet, you see).

 

[Mindscrape]

I'm debating whether or not Chris answered everything to my heart's desire, or whether it's a genius oversimplification. He covered anything I could possibly mention, but at the same time almost anything can be stripped down to its roots and simplified to a very basic level, which I don't know if I would call genius or merely a sensible check. Things are complicated, and I feel that too much reduction does an injustice rather than a service to the problem, though I do see the point that it takes a lot of prowess to take a spherical harmonic and reduce it to nothing more than polynomials (legendre polys). At the same time, you are right that circles and circular derivatives do appear in many situations, which would imply the constant pi to be appear in at least an equal number of situations.

I see the point, and it has been argued very well, much better than I could possible do, but at the same time why should heat conduction care whether it follows a circular function or hexagonal one? Is it a coincidence that physical phenomena all like to oscillate and all like to relate to circular functions (why not oscillate triangularly?), something that has been determined by a higher power, or just something we are doing to make ourselves happy with our mathematical invention? Sure a pendulum follows a circular path, with small angles of course, but why should an atom follow the same principles?

 

[matt grime]

As Chris eloquently points out, we have chosen trig functions as our preferred bases in so many situations that it is no great mystery that we see pi. Had we chosen other functions, we wouldn't see pi, or some multiple of a power of pi appearing, but entirely 'different' constants.

 

[ZapperZ]

I see the point, and it has been argued very well, much better than I could possible do, but at the same time why should heat conduction care whether it follows a circular function or hexagonal one? Is it a coincidence that physical phenomena all like to oscillate and all like to relate to circular functions (why not oscillate triangularly?), something that has been determined by a higher power, or just something we are doing to make ourselves happy with our mathematical invention? Sure a pendulum follows a circular path, with small angles of course, but why should an atom follow the same principles?

It is because every physical problem that you deal with involves the GEOMETRY of the system. A physical pendulum follows a particular geometry whether you choose a cartesian or polar coordinates. An atom has to be solve in real and momentum space when you write down the Schrodinger equation. A waveguide would have to be solved using Poisson's equation that require the geometry of the fields and the boundary conditions.

The way you are asking is similar to someone asking why was it possible that I used a screw driver to work on my electronics, yet I also need a screw driver to build a house, or a screw driver to repair a toy. Was there some coincidence or a "higher power" involved here to make that screw driver applicable to all these different areas? It is the same with mathematics.

Zz.

 

[RetardedBastard]

The way you are asking is similar to someone asking why was it possible that I used a screw driver to work on my electronics, yet I also need a screw driver to build a house, or a screw driver to repair a toy. Was there some coincidence or a "higher power" involved here to make that screw driver applicable to all these different areas?

Zz.

Yes, there WAS a higher intelligent power -- Henry F. Phillips :)

 

[ZapperZ]

Yes, there WAS a higher intelligent power -- Henry F. Phillips :)

But I wasn't using a Phillips screw driver!

:)

Zz.

 

[tony873004]

The number of seconds in a year is coincidentally similar to pi: 3.15*10⁷ seconds. I had an Astronomy teacher who loved to do "order of magnitude" solutions, and in problems that contained both pi, and seconds in a year, she would cancel out pi and the 3.15, leaving only 10⁷. It was justified since some of the inputs to the problem were expressed to fewer than 2 significant figures.

 

[Cexy]

A fleeting and whimsical thought: Could the propensity of pi to appear in mathematical formulae be explained by a kind of Zipf's law? That is, it would be especially surprising if there was no number which turned up more than any other. Given that we can expect some numbers to turn up more than others, we shouldn't be surprised about the exact form of that number - pi in this case.

It's like the philosophically-minded golfer who, reasoning that the probability of his landing his golf ball on any particular blade of grass is practically zero, congratulates himself on being a true pro when he takes a swing and finds that he does indeed manage to land his ball exactly on a blade of grass.

 

[alphachapmtl]

"...it is strange how much pi, the ratio of circumference to diameter, occurs in so much that has nothing to do with circles..."

It's true, but at the same time if you dig a bit you will find a circle everywhere.

Example: How many ways for two integer square to sum to n ?
That is how many solutions of a^2+b^2=n ? Answer=PI on average.
Unexpected at first, but a^2+b^2=n is just the equation of a circle of radius n.

Of course, sometimes its not so clear.

 

[Count Iblis]

The number of seconds in a year is coincidentally similar to pi: 3.15*10⁷ seconds. I had an Astronomy teacher who loved to do "order of magnitude" solutions, and in problems that contained both pi, and seconds in a year, she would cancel out pi and the 3.15, leaving only 10⁷. It was justified since some of the inputs to the problem were expressed to fewer than 2 significant figures.

Pi in the Sky :rofl:

 

[Gokul43201]

Pi in the Sky :rofl:

:rofl:

As opposed to $e^{i \pi}$, which is pi in your phase !

 

[Mindscrape]

Those jokes were pious.

 

[alphachapmtl]

Enough said, Pie Pie for now...

 

[robert Ihnot]

ice109: you want a really weird occurrence of pi? look up some of the series that sum to pi. e.g. $$\pi = \frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots$$

While what follows seems like baby steps in compairson with what others have written, especially about Physics, what is at stake in the above is The Leibniz Formula, which comes out of:

$$\frac{1}{1+x^2} =1-x^2+x^4-x^6-+-...$$ which strictly speaking seems only good for absolute value of x less than 1, but can be extended in this case.

Integrating both sides we arrive at pi/4 = arctan(1) = the Leibniz series.

Now tracing it back a little more we have the question of the integral $$\int_0^1 \frac{1}{x^2+1}$$ which depends upon differentation, which depends upon sin(x)/x goes to 1 as X goes to zero. And the relationship:

sec^2 =tan^2+1. This depends upon s^2+cos^2 =1, which turns on the Pythagorean theorem.

So, other than dealing with infinite series and the Pythagorean theorem, the whole matter above of pi then can be traced back to a limit observation of the arc and the sin of a smaller and smaller angle. The arc, of course, involves radians, and so that seems the key factor.

alphachapmtL "...it is strange how much pi, the ratio of circumference to diameter, occurs in so much that has nothing to do with circles..."

But in the above problem we see it DOES deal with an arc of a circle and that seems true anytime we involve the sine and are involved in differentation!

 

[Gib Z]

The series you stated is actually quite interesting, as it yields the 'would be' sum of Grandi's series: $$\sum_{n=0}^{\infty} (-1)^n$$, which we all know that by the sum of partial sums definition, diverges (or oscillates between the accumulation points 0 and 1), however many summation methods such as Cesaro sum yield the sum to be 1/2. And letting x=1 in the case of the Leibniz series gives exactly that result.

I don't really see the value of the statement that the above series for pi is a result of the Pythagorean theorem, when definitions of sin and cos in terms of taylor series, solutions to differential equations, or in terms of the exponential function all yield that same identity.

For most results one can go as far back as they want to start from the most basic result. Cauchy's generalised mean value theorem from the generic version, which is usually proved from the less general Roelle's theorem, which comes off the simple idea that between a given closed interval, a continuous real function must attain a minimum and maximum value at some point in the interval.

In other words, Cauchy's generalized mean value theorem; That for continuous real functions f(t) and g(t) over the closed interval [a,b], there exists some number c such that the following holds: $$\frac {f'(c)} {g'(c)} = \frac {f(b) - f(a)} {g(b) - g(a)}$$, depends solely on the fact that real functions that are continuous over the interval [a,b] must attain some maximum and some minimum value between that interval.

 

[robert Ihnot]

Well, starting from the Taylor series depends generally on differentation, unless you pull the definition out of thin air. We are not allowed to start with differentation in my book. (In fact the Leibnz series was not found before the Calculus was worked on by Leibnz. I am historically correct.) Where does the f'(c) come from? The idea of the max and min, and where did you get that?

From the Pythagorean theorem, we get that cos^2 + sin^2 =1. We are going to need that, and then sec^2 = tan^2 +1. Once we have the derivative of the sin, we can proceed. We start with y = arctan(x), thus x=tan(y) and differentiate. dy/dx = 1/sec^2(y)=1/[tan^2(y)+1]=1/(x^2+1).

Of course, my ideas are just conjecture, I guess. But they do explain to me the presence of pi. Do your ideas explain the presence of pi?

 

[Count Iblis]

$$e^{\pi\sqrt{n}}$$ is very close to an integer for many small integers n See here

 

[Gib Z]

Well, starting from the Taylor series depends generally on differentation, unless you pull the definition out of thin air. We are not allowed to start with differentation in my book. (In fact the Leibnz series was not found before the Calculus was worked on by Leibnz. I am historically correct.) Where does the f'(c) come from? The idea of the max and min, and where did you get that?

From the Pythagorean theorem, we get that cos^2 + sin^2 =1. We are going to need that, and then sec^2 = tan^2 +1. Once we have the derivative of the sin, we can proceed. We start with y = arctan(x), thus x=tan(y) and differentiate. dy/dx = 1/sec^2(y)=1/[tan^2(y)+1]=1/(x^2+1).

Of course, my ideas are just conjecture, I guess. But they do explain to me the presence of pi. Do your ideas explain the presence of pi?

I just don't see how relating that series back to the Pythagorean theorem has any significance, the theorem is the most basic result that is applied to get the series but that it all it means. My example was that even for something as deep as Cauchy's mean value theorem, we start with a very simple idea that is required to prove it. It is the same for all results, so I don't see why pointing out in this case the basic result was the Pythagorean theorem.

$$e^{\pi\sqrt{n}}$$ is very close to an integer for many small integers n See here

If you follow that link it is actually quite interesting, because the one closest to an integer is n=163, and 163 comes up a surprisingly large amount of times when studying deep properties of pi. One example I can give is that the following approximation is valid for about 70 digits:

$$\frac{\log_e (640320^3 + 744)}{\sqrt{163}}$$

As I said, the value of n for which $\exp (\pi \sqrt{n})$ is closest to an integer is given by 163. So it is no surprise that the value of n which yields the second closest to an integer is n=652. Why? Because [itex]\sqrt{652}=\sqrt{4\cdot163}=2\sqrt{163}. Whats the square of an almost integer? Another almost integer =]

A prize will be awarded to anyone who can either convincingly argue that this is coincidence, or who can explain why this is so in terms intelligible to an intelligent college senior.

If a college senior can understand complex multiplication and elliptic curves, then it is possible for some values of n. For others, especially the ones with the largest deviation from an integer, it is really just coincidence.

 

[robert Ihnot]

GibZIt is the same for all results, so I don't see why pointing out in this case the basic result was the Pythagorean theorem.

I did not mean that, the important sentence is:

So, other than dealing with infinite series and the Pythagorean theorem, the whole matter above of pi then can be traced back to a limit observation of the arc and the sin of a smaller and smaller angle. The arc, of course, involves radians, and so that seems the key factor.

 

[Gib Z]

How does it related to the observation of the arc and sin of a smaller and smaller angle? Did you mean the tan of 1? But yes that is correct, involving radians will bring pi into the matter, naturally.

 

[robert Ihnot]

GibZow does it related to the observation of the arc and sin of a smaller and smaller angle? Did you mean the tan of 1? But yes that is correct, involving radians will bring pi into the matter, naturally

I am talking about the derivative of the sine, which is the cosine, we have to find $$lim \frac{sinh}{h}\rightarrow1$$ $$h\rightarrow0$$ This can be found in almost any Calculus book.

It's when we get to the Calculus that we have to use radians. And it's only at the time of the Calculus that the Leibniz found his formula.

The original definition of the sine is the side opposite over the hypotenuse. But in the Calculus we are using, what can be called circular trigonometric functions, where the angle is defined in terms of the unit circle.

 

[Gib Z]

O ok see what you were talking about now. Thank you for your patience and help Robert.

 

[robert Ihnot]

Your welcome!

 

[jwalker1196]

Is it possible, given the infinite possible "base" number systems that can be used (we use base 10 of course) that e and pi have a very simple definition in one "native" base system?

This lends itself to a creator of course. Or an influencer. ZapperZ made a very good analogy.

I myself am a firm agnostic... but that doesn't mean there isn't something out there. We just haven't found it yet, so I'll wait.

 

[Gib Z]

I don't see the problem, pi and e have the same definitions in any base, and the aren't extremely complex anyway.

In any base e is the unique number that makes the integral true: $$\int^e_1 \frac{1}{dx} dx = 1$$ IE e is the unique number where the area under the graph of 1/x from 1 to e is equal to 1.

Also, in any base, pi units is half the length of the circumference of the unit circle, (though only in a Euclidean space), or the smallest positive value of x for which sin (x) is equal to 0.