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:

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