How many different representations of pi can we come up with? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:26:21Z http://mathoverflow.net/feeds/question/8825 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with How many different representations of pi can we come up with? Alex Basson 2009-12-14T02:13:05Z 2010-08-29T06:07:09Z <p>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:</p> <ol> <li>It has to be mathematically correct.</li> <li>It has to be either a representation of pi itself or lead directly to a representation of pi.</li> </ol> <p>So for example, this is okay: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ (because it equals $\frac{\pi^2}{6}$)<br /> But this is not: $\frac{22}{7}$.</p> <p>How many can we come up with?</p> http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with/8827#8827 Answer by Harrison Brown for How many different representations of pi can we come up with? Harrison Brown 2009-12-14T02:45:07Z 2009-12-14T02:45:07Z <p>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.</p> <p>One thing that Mathworld doesn't list that could be interesting is <a href="http://en.wikipedia.org/wiki/Buffon%2527s%5Fneedle" rel="nofollow">Buffon's needle</a>. 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.</p> http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with/8828#8828 Answer by Sam Nead for How many different representations of pi can we come up with? Sam Nead 2009-12-14T02:46:40Z 2009-12-14T02:46:40Z <p>I was reminded of this classic a few days ago: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots$</p> <p>Is there a slick proof of this (in particular one that avoids discussing arctan for a long time)? </p> http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with/8834#8834 Answer by Steve Huntsman for How many different representations of pi can we come up with? Steve Huntsman 2009-12-14T03:30:08Z 2009-12-14T03:30:08Z <p>Hexadecimal expansion of $\pi$: <a href="http://mathworld.wolfram.com/BBPFormula.html" rel="nofollow">http://mathworld.wolfram.com/BBPFormula.html</a></p> http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with/8953#8953 Answer by Christian Bjartli for How many different representations of pi can we come up with? Christian Bjartli 2009-12-15T04:27:14Z 2009-12-15T04:27:14Z <p>Let me suggest getting a striped floor and tiling it with randomly placed needles. </p> http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with/36959#36959 Answer by mark for How many different representations of pi can we come up with? mark 2010-08-28T09:04:18Z 2010-08-28T09:04:18Z <p>check this out <a href="http://www.telegraph.co.uk/news/newstopics/howaboutthat/2144652/Most-complex-crop-circle-ever-discovered-in-British-fields.html" rel="nofollow">http://www.telegraph.co.uk/news/newstopics/howaboutthat/2144652/Most-complex-crop-circle-ever-discovered-in-British-fields.html</a></p> http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with/36990#36990 Answer by Bernikov for How many different representations of pi can we come up with? Bernikov 2010-08-28T19:39:38Z 2010-08-28T19:39:38Z <p>I really like this formula: $1+\frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \frac{1}{1\cdot 3\cdot 5\cdot 7} + \frac{1}{1\cdot 3\cdot 5\cdot 7\cdot 9} + \cdots + {{1\over 1 + {1\over 1 + {2\over 1 + {3\over 1 + {4\over 1 + {5\over 1 + \cdots }}}}}}} = \sqrt{\frac{e\cdot\pi}{2}}$</p> http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with/37026#37026 Answer by Gerry Myerson for How many different representations of pi can we come up with? Gerry Myerson 2010-08-29T06:07:09Z 2010-08-29T06:07:09Z <p>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. </p>