Question

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?

Answer 1

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.

Answer 2

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

Answer 3

check this out http://www.telegraph.co.uk/news/newstopics/howaboutthat/2144652/Most-complex-crop-circle-ever-discovered-in-British-fields.html

Answer 4

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

Answer 5

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

Answer 6

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)?

Answer 7

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.