Pi appears a LOT in trigonometry, but only because of its 'circle-significance'. Does pi ever matter in things not concerned with circles? Is its only claim to fame the fact that its irrational and an important ratio?
Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference?
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2I'd rather this be a comment instead, so: $\pi$ turns up in the expression for the so-called "probability integral" (a.k.a. the "error function") among other things. How circles relate to this is a bit of a long-winded explanation though. – 2010-08-26
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4Have you read http://en.wikipedia.org/wiki/Pi#Use_in_mathematics_and_science ? – 2010-08-26
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16Also, let's get one thing straight here: circles are _eerily important._ You will never stop running into circles in mathematics. – 2010-08-26
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5(For example, although the Fourier transform is "concerned with circles" (functions on the circle being the same thing as periodic functions) it penetrates into the deepest parts of modern mathematics. Many appearances of pi are because of a Fourier transform lurking somewhere in the background. You might also want to read this MO thread where I asked a similar question: http://mathoverflow.net/questions/18180/what-are-some-fundamental-sources-for-the-appearance-of-pi-in-mathematics) – 2010-08-26
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0Qiaochu: good that you linked to it; I like gowers's "circles and rotations appear a lot." – 2010-08-26
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4Fundamental source of $\pi$ is circle nothing else. It may be difficult to find it but it is always there. – 2010-08-26
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0are any of these places simplified by replacing π with τ/2? http://tauday.com/ – 2011-03-23
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0Since you can define $\pi$ to be the ratio of a circle's circumference to its diameter, you can always reverse-engineer some context in which $\pi$ appears to get something to do with circles. **Proof:** Suppose we have some interesting fact about $\pi$, and let $P$ be a totally self-contained proof of that fact. Then there must be a first statement $s$ in $P$ at which the number $\pi$ appears. Now suppose we have defined $\pi$ to be the ratio of a circle's circumference to its diameter. But then in order to introduce $\pi$ into – 2012-04-23
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0(contd.) our proof we have to work straight from the definition of $\pi$ (since $P$ is self contained). So we have to introduce circles into our proof at some point before $p$. – 2012-04-23
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0Hmmm.. I seem to have allowed for the possibility that there exists some statement about $\pi$ which is true but which is not provable in any order of logic, and which has nothing to do with circles. Now that is an interesting thought. – 2012-04-23
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0On a related note: http://www.smbc-comics.com/?db=comics&id=2420#comic – 2012-04-23
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0Often convergence includes PI. For example, [Hardy's proof of the Dirichlet eta function](https://en.wikipedia.org/wiki/Dirichlet_eta_function) – 2012-10-18
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0Sometimes it seems one will never stop running in circles either. :-) – 2013-06-09
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0http://numbers.computation.free.fr/Constants/Pi/piSeries.html – 2013-10-27