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The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!]
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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 .

   

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 .

   

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
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The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] - by jaydfox - 09/07/2007, 06:42 PM

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