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It's a fact that in many physical formulae $\pi$ (or even multiples of it as far as I can see) and $e$ show up. But why would that be so? Is because the two are "connected" in the well-known formula $e^{i\pi}+1=0$, though this only shows the two in one formula and, I think, can't be seen as an answer because it only shifts the problem.

Also in statistics, they show up: For example, the chance that the needle in Buffon's needle problem crosses one of the lines is $\frac{2}{\pi}$. Also, $e$ shows up in many statistical formulae.

Another example (this time for $e$, which is not addressed in the question of which this question is supposed to be a duplicate) is the use of $e$ in the calculation of exponential decay rates.

Is there something "deep" about these two numbers?

P.S. Also, the golden ratio appears in many physical phenomena, as expressed in the many forms that appear in Nature and which has nothing to do (in the case of $\pi$) with spherical symmetry, the volume of n-spheres, trigonometric functions, Fourier analysis, etc. Neither with $e$, so also of $\varphi$ can be said it's another example of one of the most fundamental numbers.

I think we overlook many aspects of Nature if at the heart of mathematics, as suggested by the answer, lay $\pi$ in geometry and $e$ in calculus. Aren't there many areas of math (or math that still has to be developed) in which these numbers don't form "the fundamental heart", but nevertheless can be applied in investigating Nature mathematically? In this case, the answer to my question is likely that the number of formulae which contain $\pi$ or $e$ would be that in fact, the relative number of these formulae ain't that big (though the absolute number is).

Deschele Schilder
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  • "Is there something "deep" about these two numbers?" - see Logarithmic spiral – Alfred Centauri Dec 13 '17 at 03:50
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    Our of curiosity, what physics formulae have a $\pi$ in it? Most of the ones I can think of have $\pi$ because there's some kind of frequency present, and then there are examples where the surface area of a sphere comes into play. – DanielSank Dec 13 '17 at 05:36
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    $\pi$ aspect is an exact dupe of this earlier question. – Kyle Kanos Dec 13 '17 at 12:40
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    @DanielSank I think what you say is very interesting. I have searched for the character $\pi$ in the file https://physics.illinois.edu/academics/graduates/physics-formulary.pdf and found 233 of them. I skimmed through all and as far as I can see you are right. I think whenever there is a $4\pi$ it is some kind of integration in spherical coordinates and as soon as there is a $2\pi$ it is frequency related. Interesting. – physicopath Dec 13 '17 at 13:12
  • @physicopath Yes, this is why I find probably_someone's answer a bit unsatisfying. Unfortunately, this post was marked as a duplicate so I can't offer my own answer. I'm tempted to edit this post a bit to make it not a duplicate... – DanielSank Dec 14 '17 at 03:19
  • @DanielSank-If you want to, go ahead!! I totally agree with you. – Deschele Schilder Dec 14 '17 at 05:58
  • You can also choose to reopen the question (which you maybe already did). – Deschele Schilder Dec 14 '17 at 08:29

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In physics, geometry and calculus show up almost everywhere. At the heart of geometry is $\pi$, and at the heart of calculus is $e$. I don't see how it's at all surprising that two of the most fundamentally important numbers in mathematics tend to show up a lot when mathematics is used.

probably_someone
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    Maybe the question could be rephrased as: why does $\pi$ show up even when there are no circles in sight? – Jahan Claes Dec 13 '17 at 03:32
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    @JahanClaes That question has many answers. For one, many things revolve in circles, or rotate (in which all of their internal points revolve in circles). Still other things move sinusoidally, and a sinusoid is just an object moving in a circle as viewed from the edge. Spherical symmetry also shows up whenever interactions don't care about the direction they're coming from. – probably_someone Dec 13 '17 at 03:35
  • probably_someone is right- anytime you have something that lives in 3D space, you get pi popping up in places you wouldn't expect it- there is a wonderful little book about pi (can't remember the author) that is full of examples of this. – niels nielsen Dec 13 '17 at 05:13
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    Sure $\pi$ may be common in trigonometry, but claiming that it's at the heart of geometry is kind of reductionist. Similarly, I'm not sure I'd put $e$ at the heart of calculus. The exponential function is special because it's an eigenvector of the derivative and the derivative is at the heart of calculus, but I think that fact should be related to physics by explaining how derivatives are related to symmetry, etc. – DanielSank Dec 13 '17 at 05:30
  • @DanielSank That all depends on how much detail you want to get into. The above is easy to explain to a potential layperson without having to go too much into abstraction (which in my experience causes people's eyes to glaze over). If the OP requests more detail, I'll be happy to add it. – probably_someone Dec 13 '17 at 05:34
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    @probably_someone I understand what you mean and I appreciate that approach. At the same time, I'm not sure it's a question of detail. Arguing physics $\to$ math $\to e,\pi$ sounds simple, but it doesn't give the reader an understanding of why $\pi$ and $e$ show up in Nature. Pointing out a few examples where $\pi$ shows up in a formula and explaining what physical thing leads to the presence of that $\pi$ would be a nice improvement, in my opinion. – DanielSank Dec 13 '17 at 05:40
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    Read the OP's profile - I think he would like some more detail. He's studied group theory, and has a physics degree. – Selene Routley Dec 13 '17 at 06:23
  • @probably_someone- Why are $\pi$ and $e$ the most fundamentally important numbers in mathematics? Why isn't, for example, the golden number (1.68...) less important. It seems that the golden ratio plays a big role in Nature, where it is to find in the many forms Nature cooks up. – Deschele Schilder Dec 13 '17 at 13:30
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    It might be worth pointing out Poisson's equation, which frequently gets a $4\pi$ in either the source term or the solution — for reasons of spherical dependence — and complex analysis, which frequently results in $2\pi$ due to Cauchy's formula. – Mike Dec 13 '17 at 22:00
  • @probably_someone-I think this answer is way too short! – Deschele Schilder Dec 14 '17 at 06:56