## How many different representations of pi can we come up with?

Let me explain: a friend of a friend is opening a new pizza restaurant called "Pi", and he's looking to decorate his walls with pi-related material: formulas, equations, theorems w/ proof, diagrams, etc. Any suggestion is welcome, so long as it meets these two criteria:

1. It has to be mathematically correct.
2. It has to be either a representation of pi itself or lead directly to a representation of pi.

So for example, this is okay: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ (because it equals $\frac{\pi^2}{6}$)
But this is not: $\frac{22}{7}$.

How many can we come up with?

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There are 158 of them listed [here](functions.wolfram.com/Constants/Pi)... – Mariano Suárez-Alvarez Dec 14 2009 at 2:14
I'm not totally convinced that this is appropriate for MO, and I imagine it'd be less controversial at someplace like Art of Problem Solving. But at a bare minimum this should be community wiki. – Harrison Brown Dec 14 2009 at 2:15
Well, it's not 158... the list includes things like the formula for the absolute value of $\pi$ (!) – Mariano Suárez-Alvarez Dec 14 2009 at 2:29
This sounds like a fun question, that could perhaps (slightly) help somebody out in the real world. +1 – Sam Nead Dec 14 2009 at 2:38

Let me suggest getting a striped floor and tiling it with randomly placed needles.

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 Try explaining that to the health inspectors... – Cam McLeman Aug 28 2010 at 21:56

Hexadecimal expansion of $\pi$: http://mathworld.wolfram.com/BBPFormula.html

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Well, Mathworld does list a bunch of formulas. Some of them are more mathematically interesting than others (BBP, Ramanujan's, $\zeta(2)$), but purely aesthetically they're probably all about the same.

One thing that Mathworld doesn't list that could be interesting is Buffon's needle. I'm not sure how you'd represent it statically, but it's easy to understand and surprising, which is a good combination for a lay audience.

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 Here are links to Pi Formulas. – Douglas Zare Jan 13 2010 at 8:48

I really like this formula: $1+\frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \frac{1}{1\cdot 3\cdot 5\cdot 7} + \frac{1}{1\cdot 3\cdot 5\cdot 7\cdot 9} + \cdots + {{1\over 1 + {1\over 1 + {2\over 1 + {3\over 1 + {4\over 1 + {5\over 1 + \cdots }}}}}}} = \sqrt{\frac{e\cdot\pi}{2}}$

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I was reminded of this classic a few days ago: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots$

Is there a slick proof of this (in particular one that avoids discussing arctan for a long time)?

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Slicker that this <en.wikipedia.org/wiki/…;? – Mariano Suárez-Alvarez Dec 14 2009 at 3:02
That works - I didn't click through far enough in Wikipedia when I went to en.wikipedia.org/wiki/Pi#History Thanks for the link! – Sam Nead Dec 14 2009 at 3:49
Beware! <xkcd.com/214/>; :) – Mariano Suárez-Alvarez Dec 15 2009 at 3:38

Chapter 16 of Jorg Arndt and Christoph Haenel, $\pi$ Unleashed, is called $\pi$ Formula Collection, and has well over 100 formulas. Another source is Pierre Eymard and Jean-Pierre Lafon, The Number $\pi$, published by the American Math Society.

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 Not really what the original question was asking for – Yemon Choi Aug 28 2010 at 20:00