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 The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 09/07/2007, 06:42 PM (This post was last modified: 09/07/2007, 07:00 PM by jaydfox.) The following graphs show successive iterations of the natural logarithm of the real unit interval (0, 1). The first graph shows the second iteration, which returns a straight line with endpoints $+\infty + \pi i, -\infty + \pi i$.     The second graph shows the third iteration, which is a sort of trigonometric curve (real part is logarithm of cosecant, I think) with endpoints at $+\infty, \infty + \pi i$.     The iterations after that don't have as symmetric a structure. But they sure are fascinating. Here's the fourth iteration (leaving the previous iterations in place for reference):     Please note that all the successive iterations after the third will snake between the previous and next iterations to reach endpoints at positive infinity. Here's the fifth and sixth iterations:         ~ Jay Daniel Fox « Next Oldest | Next Newest »

 Messages In This Thread The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/07/2007, 06:42 PM RE: The fractal nature of iterated natural logarithms - by jaydfox - 09/07/2007, 06:46 PM RE: The fractal nature of iterated natural logarithms - by jaydfox - 09/07/2007, 06:56 PM RE: The fractal nature of iterated natural logarithms - by Gottfried - 09/07/2007, 08:15 PM RE: The fractal nature of iterated natural logarithms - by jaydfox - 09/07/2007, 08:31 PM RE: The fractal nature of iterated natural logarithms - by jaydfox - 09/07/2007, 06:59 PM RE: The fractal nature of iterated natural logarithms - by bo198214 - 09/07/2007, 07:44 PM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/07/2007, 08:21 PM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/07/2007, 09:13 PM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/07/2007, 10:47 PM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by bo198214 - 09/08/2007, 05:06 PM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/08/2007, 05:39 PM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/08/2007, 05:15 AM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by Gottfried - 09/08/2007, 08:43 AM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/08/2007, 05:46 PM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by bo198214 - 09/08/2007, 07:03 PM RE: The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/09/2007, 01:21 AM

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