# Pi in everything

1. Sep 26, 2007

### 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."

2. Sep 26, 2007

### 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.

3. Sep 26, 2007

### ice109

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.

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

4. Sep 26, 2007

### 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...

5. Sep 26, 2007

### mjsd

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.

6. Sep 26, 2007

### HallsofIvy

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.

7. Sep 26, 2007

### 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!

Last edited: Sep 26, 2007
8. Sep 26, 2007

### HallsofIvy

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"!

9. Sep 26, 2007

### Chris Hillman

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

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.)

Last edited: Sep 26, 2007
10. Sep 26, 2007

### ice109

chris hillman who are you!

11. Sep 26, 2007

### matt grime

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

12. Sep 26, 2007

### 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?

13. Sep 27, 2007

### 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.

14. Sep 27, 2007

### ZapperZ

Staff Emeritus
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.

15. Sep 27, 2007

### RetardedBastard

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

16. Sep 27, 2007

### ZapperZ

Staff Emeritus
But I wasn't using a Phillips screw driver!

:)

Zz.

17. Oct 3, 2007

### 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.

18. Oct 9, 2007

### 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.

19. Oct 10, 2007

### 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.

20. Oct 10, 2007

### Count Iblis

Pi in the Sky :rofl:

21. Oct 10, 2007

### Gokul43201

Staff Emeritus
:rofl:

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

22. Oct 11, 2007

### Mindscrape

Those jokes were pious.

23. Oct 12, 2007

### alphachapmtl

Enough said, Pie Pie for now...

24. Oct 12, 2007

### 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!

Last edited: Oct 12, 2007
25. Oct 12, 2007

### 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.