{"id":581,"date":"2008-08-31T16:00:43","date_gmt":"2008-08-31T23:00:43","guid":{"rendered":"https:\/\/www.reenigne.org\/blog\/?p=581"},"modified":"2011-08-08T15:54:55","modified_gmt":"2011-08-08T22:54:55","slug":"fourier-transform-of-the-mandelbrot-set","status":"publish","type":"post","link":"https:\/\/www.reenigne.org\/blog\/fourier-transform-of-the-mandelbrot-set\/","title":{"rendered":"Fourier transform of the Mandelbrot set"},"content":{"rendered":"<p>I wonder what the Fourier transform of the Mandelbrot set looks like? More specifically, the 2D Fourier transform of the function f(x,y) = {0 if x+iy is in M, 1 otherwise}. This has infinitely fine features, so the Fourier transform will extend out infinitely far from the origin. It's aperiodic, so the Fourier transform will be non-discrete.<\/p>\n<p>The result will be a complex-valued function of complex numbers (since each point in the frequency domain has a phase and amplitude). That raises the question of its analytical properties - is it analytic everywhere, in some places or nowhere? (Probably nowhere).<\/p>\n<p>Other interesting Mandelbrot-set related functions that could also be Fourier transformed:<br \/>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=M_n%28x%2Cy%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='M_n(x,y)' title='M_n(x,y)' class='latex' \/> = the nth iterate of the Mandelbrot equation (<img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+f+%3D+%7Ce%5E%7B-%5Clim_%7Bn+%5Cto+%5Cinfty%7D%5Cfrac%7B1%7D%7Bn%7DM_n%7D%7C&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\displaystyle f = |e^{-\\lim_{n \\to \\infty}\\frac{1}{n}M_n}|' title='\\displaystyle f = |e^{-\\lim_{n \\to \\infty}\\frac{1}{n}M_n}|' class='latex' \/>).<br \/>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=D%28x%2Cy%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='D(x,y)' title='D(x,y)' class='latex' \/> = distance between x+iy and the closest point in the Mandelbrot set. Phase could also encode direction.<br \/>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=P%28x%2Cy%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='P(x,y)' title='P(x,y)' class='latex' \/> = the potential field around an electrically charged Mandelbrot set.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I wonder what the Fourier transform of the Mandelbrot set looks like? More specifically, the 2D Fourier transform of the function f(x,y) = {0 if x+iy is in M, 1 otherwise}. This has infinitely fine features, so the Fourier transform will extend out infinitely far from the origin. It's aperiodic, so the Fourier transform will [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18],"tags":[],"class_list":["post-581","post","type-post","status-publish","format-standard","hentry","category-fractals"],"_links":{"self":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/581","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/comments?post=581"}],"version-history":[{"count":4,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/581\/revisions"}],"predecessor-version":[{"id":585,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/581\/revisions\/585"}],"wp:attachment":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/media?parent=581"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/categories?post=581"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/tags?post=581"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}