Index of /~peter/projects/math/fractal

 Name                    Last modified      Size  Description
 Parent Directory                             -   
 README.html             20-Aug-2003 13:51  1.8K  
 bigzoom.gif             11-Oct-2004 23:38  278K  
 doit.sh                 28-Jul-2003 14:25  191   
 frac.py                 18-Jul-2003 09:47  5.4K  
 frac.pyc                17-Jul-2003 17:40  4.1K  
 fractal.desc            02-Feb-2004 23:54  943   
 smart_frac.py           24-Jul-2003 14:49  5.4K  
 thing.gif               11-Oct-2004 23:42   17K  
 tiny.png                18-Jul-2003 10:23  1.0K  
 zoom.gif                11-Oct-2004 23:41  178K  

A mathematical fun fact is that the equation xxxxxxxxx...=2 has a solution of x=sqrt(2). Well, when does this equation diverge, and when does it converge?

UPDATE! Answer found:

Baker and Rippon, "Convergence of infinite exponentials" Ann Acad Sci Fenn Ser AI Math 1983 179-186 showed that z^zz... converges for log(z) contained in {t*e^-t: |t| < 1 or tⁿ = 1 for some n= 1, 2, ..} and that it diverges elsewhere.(from here and here)

I thought I found a new fractal, so I investigated the hell out of it. It's nice to know that I was on the forefront of some pretty interesting mathematics 20 years ago.

It turns out that the answer to that questions is not nearly as clear cut as I wish it were. As an example, -.15 doesn't seem to diverge, but I'm pretty sure that -.9 does, even though -1 doesn't, and -1.1 does. And -1.75 doesn't. Seriously. What the hell? And when you throw the entirety of the complex plane into it, the picture looks even more confusing. Some values seem to be cyclic - i.e. a tower of x's that is of height 5 mod 6 will always have the same value, but that value will be different from a tower of height 2 mod 6. Time for me to get a geometric interpretation of complex exponentiation.

Here is a tiny picture of it:

For more info, try making them yourself with the program frac.py > filename.pgm .