03/01/2012, 10:34 AM
(This post was last modified: 03/01/2012, 10:42 AM by sheldonison.)

Here are some plots, as the base loops around eta. The first image is for base=eta+0.25, which is tetration at the real axis. Then each subsequent image is for the base looped around eta, pi/12th of a circle, counter clockwise. Then the 12th image is for base=1.195, and then I continued around to imag(z)<0. I stopped at B=1.621-0.1768i, which didn't converge. I included it anyway, in the image, just to show what might happens. My guess is that these merged solutions cannot be continued past the 2nd Shell Thron boundary, even though they can be continued from tetration at the real axis, to the first Shell Thron boundary in the upper half of the complex plane. But continuing around to bases less than eta, and than to bases with imag(z)<0, solutions become impossible the second time you cross the Shell Thron boundary. The grid lines for all of these plots are at 5 units, with the center of the plot at z=0, where sexp(z)=1.

- Sheldon

This is the first 8 bases, looping counter clockwise around eta, starting with eta+0.25. The first four bases are on the left, and the next four bases continue on the right.

This is the next 8 bases, continuing counterclockwise around eta. The left four images are in the upper half of the complex plane. The right four images continue, starting at the real axis for eta-0.25, and then rotate into the lower half of the complex plane.

Continuing on, in the lower half of the complex plane.

- Sheldon

This is the first 8 bases, looping counter clockwise around eta, starting with eta+0.25. The first four bases are on the left, and the next four bases continue on the right.

This is the next 8 bases, continuing counterclockwise around eta. The left four images are in the upper half of the complex plane. The right four images continue, starting at the real axis for eta-0.25, and then rotate into the lower half of the complex plane.

Continuing on, in the lower half of the complex plane.