Hello again.

That's my first entry within the new forum.

As usual not much time etc. - But this finding was promising enough to share it :-)

//NOTE:

* I don't see a really matching forum chapter - This entry is not only for MB3D, nor a pure image thread, not real programming, not real theory. *

If anyone with admin rights feels that it should be moved - please do so :-)The purpose of this thread is to bring up an old topic again - a 3D representation of the good old Newton fractal.

This is program independent - In fact I still don't see a good (or any) realization in the most 3D fractal programs.

What we currently have appears to be still like in the 2 years old thread

http://www.fractalforums.com/gallery/newton-3d/ (But please feel free to tell if I have missed something!)

But when I see my pictures I think that it would be worth to implement it in any 3D fractal program!

Ok, let's become technically ---

The old question was "What the hell is the outside in a Newton - everything converges".

My "new" idea is very simple:

If I know one of the solutions (let's call it s) of t^k-c=0 then 1/(t(n) - s) would diverge, while any other solution would still converge (t(n) is the Newton method's result after n iterations).

Hence we would have a (diverging) outside, and a (converging) inside.

AND - 1/x is of course reversible. - That is important to implement such a solution in programs (like MB3D) which has built-in DE calculation, and which can only take one triplex value from one iteration into the other one.

Hence we have this basic calculation scheme (t is the current triplex number):

`(For each iteration > 1) t := 1/t+s`

Calculate the Newton for t

t := 1/(t-s)

Then the t(n) visible for the program would diverge if the "real" t is close to s, but converge else.

Of course - users of Fragemtarium etc could write their own DE check without such hacks ;-)

Ok, and here is the result of my implementation in MB3D:

1. The most simple version.

At the begin I was not aware about a possibility to check the number of the current iteration in MB3D.

In result I needed to run the pre-inversion also for the first iteration -

With the effect that the endless Newton fractal turned into a finite object -

**a Newton bulb** Better than I have have expected!

Ok, the complete "bulb" does not look too fancy, but some fruits need to be cutted or peeled ;-)

This (and nothing else) is realized in the MB3D JIT formula JIT_gnj_RealPowNewton_01.m3f

Note regarding the performance in MB3D:

The cutted fruit (picture Newton-1a) needed for a 1600x1200 picture, and a raystep of 0.5 about 35 seconds (on a 6-core i7).

**I guess that's quite fast ** for a JIT that uses exponents (real power based calculation)....

2. A tiny variation.

For testing purposes I had added some fix values before any other calculation.

I found the results quite nice - so I left it from the version 2 of my formula on (JIT_gnj_RealPowNewton_02.m3f)

Picture Newton-2c is a hybrid of 2 formulas with different powers (3 and 2).

3. The non-inversed version.

Finally I found a way to count the iteration also in MB3D - But use the formula carefully; it changes an internal variable, which may be used in other formula types as well.

This feature is available starting with JIT_gnj_RealPowNewton_03.m3f

Higher powers don't make a too big difference - remember that always just one solution will be taken as the outside...

I'm attaching the formulas and the parameters.

The formula doc can be found within the .m3f files.

So, that's it from me ---

**PLEASE FEEF FREE TO ADD IMPLEMENTATIONS FOR OTHER PROGRAMS !**