x↑↑x = -1

[sheldonison]

....Intuitively, I have a good understanding of Kneser's method. But turning that intuition into calculations is the hard part. I don't know how to do Riemann mappings in general.

I followed the steps outlined in the paper by Murid, Ali H. M. and Mohamed, Nurul Akmal (which I discussed in post #10 from this old thread)
...
Sadly, I've lost my Kneser construction code. Once I finish my slog library, that's my next big project, to rebuild the Kneser construction the way I did last time.

Jay,

I read and re-read those threads of yours at least 50 times I wanted sooo desperately to understand Kneser's construction. And this was after I had accidentally stumbled on my z+theta(z) approach, which I assumed was Kneser's construction, until it slowly dawned on me that it wasn't. I'm embarrassed to admit that it was many many months later before I finally understood Kneser's construction well enough to generate the mathematical equations relating it to the algorithm I used in the program I published here, that I blithely called "kneser.gp".

Here's my opinion now. Kneser's construction is useful primarily because it gives a theoretical framework to prove the existence of the Tetration solution. Numerically, it can't possibly give results of any reasonable precision, due to the really nasty singularity for the Abel function of 0. I'm amazed you got Riemann mapping results as good as you did! But, the algorithm used in the Kneser.gp program is really useful since it is probably the fastest and best converging solution we have for tetration, and it has the added bonus of having a theoretical equivalence to Kneser's Riemann mapping. The only thing is, there is this small detail that its hard to prove the algorithm actually converges..... But assuming it converges, than it is equivalent to Kneser's Riemann mapping.

One day, I need to actually publish some of this stuff in one of those Math journals or something. Thanks for your kind posts, both now, and for your incredibly well written posts in the past!