User randy qian - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:22:22Z http://mathoverflow.net/feeds/user/9569 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40268/why-is-the-gaussian-so-pervasive-in-mathematics Why is the Gaussian so pervasive in mathematics? Randy Qian 2010-09-28T05:06:52Z 2012-11-14T09:17:26Z <p>This is a heuristic question that I think was once asked by Serge Lang. The gaussian: \$e^{-x^2}\$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as the solution to the heat equation, in a very nice proof of the Atiyah-Singer index theorem etc. Is this an artifact of the techniques (such as the Fourier Transform) that people like use to deal with certain problems or is this the tip of some deeper platonic iceberg?</p> http://mathoverflow.net/questions/81960/the-dzhanibekov-effect-an-exercise-in-mechanics-or-fiction-explain-mathemati/82020#82020 Comment by Randy Qian Randy Qian 2011-11-28T03:10:46Z 2011-11-28T03:10:46Z That was great exposition; here is a nice video of the oscillation described: youtube.com/watch?NR=1&amp;v=LR5hkgfRPno http://mathoverflow.net/questions/40268/why-is-the-gaussian-so-pervasive-in-mathematics Comment by Randy Qian Randy Qian 2010-09-28T20:21:38Z 2010-09-28T20:21:38Z Makes sense, thanks for that explantation about spheres. A thought: there seems to be some similarity between the formulation of the heat equation as the limit of squares with heat flow through its edges proportional to the difference of temperature and the dropping balls through pegs by relating balls to heat flow. Maybe instead of thinking of the ball as going one way or another when meeting a peg, we can consider its expectation of going either way which would be closer to the heat flow model. http://mathoverflow.net/questions/40337/ingenuity-in-mathematics/40374#40374 Comment by Randy Qian Randy Qian 2010-09-28T19:45:36Z 2010-09-28T19:45:36Z I think an analogous proof that can be explained to artists is to tile the rectangle w/ a black and white checkerboard parallel to the rectangles with squares of width 1/2. You can then show that a rectangle having equal black and white areas (of overlapped checkerboard squares) is necessary and sufficient for determining that it has an integral side. Each rectangle with integral sides then has equal black and white squares, which combine to show that the large rectangle has the same property and thus an integral side itself. http://mathoverflow.net/questions/40268/why-is-the-gaussian-so-pervasive-in-mathematics Comment by Randy Qian Randy Qian 2010-09-28T19:28:54Z 2010-09-28T19:28:54Z Thanks for all the comments and nice examples. Another one of my very favorite uses of the Gaussian is to find the volume of the n-sphere <a href="http://planetmath.org/?op=getobj&amp;from=objects&amp;name=AreaOfTheNSphere" rel="nofollow">planetmath.org/&hellip;</a> but its appearance here seems to have more to do with the calculation got &quot;factored&quot; than some especially deep fact about spheres. Is there some obvious relation between the CLT and the heat equation? (I know that the Fourier transform gives 2 matching ODEs in those two scenarios, but is there some less heavy handed explanation?)