• March 18, 2018, 10:23:30 AM

### Author Topic:  Revisiting the 3D Newton  (Read 180 times)

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#### gannjondal

• Fractal Freshman
• Posts: 5
##### Revisiting the 3D Newton
« on: March 13, 2018, 01:17:19 AM »
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):

Code: [Select]
(For each iteration > 1)  t := 1/t+sCalculate the Newton for tt := 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 ---

« Last Edit: March 13, 2018, 09:56:15 PM by gannjondal »

#### 3DickUlus

• Fractal Fluff
• Posts: 372
##### Re: Revisiting the 3D Newton
« Reply #1 on: March 13, 2018, 01:39:48 AM »
Interesting, I don't think I've ever seen a Newton bulb, yes this would be a good place for it.
Resistance is fertile... you will be illuminated!

https://en.wikibooks.org/wiki/Fractals/fragmentarium

#### gannjondal

• Fractal Freshman
• Posts: 5
##### Re: Revisiting the 3D Newton
« Reply #2 on: March 16, 2018, 12:00:02 AM »
.... yes this would be a good place for it.

Thanks, 3DickUlus. Then we'll leave it here

#### v

• Fractal Fanatic
• Posts: 23
##### Re: Revisiting the 3D Newton
« Reply #3 on: March 16, 2018, 10:28:52 AM »
Neat idea, looks sort of like a surface of revolution with the smooth edges.  I wonder if it could be extended to grow in complexity in all degrees of freedom, not just in cross section.  Maybe the extension of newton's method to 3D (gradient descent?) or functions of two variables on the mandelbulb formula might produce something interesting

#### Ebanflo

• Fractal Freshman
• Posts: 7
##### Re: Revisiting the 3D Newton
« Reply #4 on: March 16, 2018, 08:53:52 PM »

So I'd like to have a precise understanding of the math so that I can replicate this beautiful object. I assume triplex number means the number system used to generate the Mandelbulb (is multiplication defined for this number system?). So your formula is applied to a point in space t:
Code: [Select]
t := t/(t + s) //what is s?compute the Newton for t //is this a single iteration of the root-finding method or do you iterate the method until it becomes really close to the roott := t/(t - s)And is this the fractal formula or the actual distance field?

Thanks!

#### Bill Snowzell

• Fractal Friar
• Posts: 105
##### Re: Revisiting the 3D Newton
« Reply #5 on: Yesterday at 09:42:44 AM »

So I'd like to have a precise understanding of the math so that I can replicate this beautiful object. I assume triplex number means the number system used to generate the Mandelbulb (is multiplication defined for this number system?). So your formula is applied to a point in space t:
Code: [Select]
t := t/(t + s) //what is s?compute the Newton for t //is this a single iteration of the root-finding method or do you iterate the method until it becomes really close to the roott := t/(t - s)And is this the fractal formula or the actual distance

Thanks!

I use MB3d with Linux Mint, it will run under "Wine" with no issues.

#### Ebanflo

• Fractal Freshman
• Posts: 7
##### Re: Revisiting the 3D Newton
« Reply #6 on: Yesterday at 09:56:19 AM »
i actually got it to run on Ubuntu with Wine shortly after posting.

#### gannjondal

• Fractal Freshman
• Posts: 5
##### Re: Revisiting the 3D Newton
« Reply #7 on: Yesterday at 07:41:56 PM »
Trying to answer the remaining of above questions ...

Neat idea, looks sort of like a surface of revolution with the smooth edges.  I wonder if it could be extended to grow in complexity in all degrees of freedom, not just in cross section.  Maybe the extension of newton's method to 3D (gradient descent?) or functions of two variables on the mandelbulb formula might produce something interesting

Increasing complexity is an own topic that could be discussed lengthy.
I have tried that once (for 2D Newton pictures) quite extensively.

From that experience I would say that in case of the Newton method 'simple' extensions like the Newton calculation using hypercomplex numbers, matrices etc do generate 'simple' pictures (like more Newton solutions), but keep the general pattern.
On the other hand simple mutations tend to destroy the complete picture.

Changes need to be more sublte to generate really interesting things.
Basically:  Blow up the formula calculated in one step, and then make a tiny change in the code on one corner of the calculations.
Hybridisation is one way (Make n calculations of the same formula, and then on calculation with a slightly changed formula).
Using n-dimensional numbers (be it mathematically useful or not), or take the Newton method as an operator, and apply it to iself are some other options (don't forget the well placed mistake).
Your idea to use 2 variables is also valid, and can even lead to interesting pictures without "mutation" - but some effort is needed as well (think e.g. about something like N[N[f(t,q),t],q] , where N[f(x),x] is the Newton method used for f(x), with derivative on x).
And many more. It's something for at least one own thread ;-)

We need to carefully think about what's inside, and what outside.
And also the surfaces must not be too complex, otherwise the picture becomes uninteresting....

-------

So I'd like to have a precise understanding of the math so that I can replicate this beautiful object. I assume triplex number means the number system used to generate the Mandelbulb (is multiplication defined for this number system?). So your formula is applied to a point in space t:
Code: [Select]
t := 1/t + s //what is s?compute the Newton for t //is this a single iteration of the root-finding method or do you iterate the method until it becomes really close to the roott := t/(t - s)And is this the fractal formula or the actual distance field?

Thanks!

But I want to try to answer your technical questions in a short way:

- I assume triplex number means the number system used to generate the Mandelbulb (is multiplication defined for this number system?)
Yes, it's the same kind of numbers. In the old forums there have been lengthy discussions about the rules we are using.
Finally they came down to some "practical useable" calculation rules that include rules for multiplication, and even division.
- t := 1/t + s //what is s? ---- t := t/(t - s) (I have corrected the equation):
s is one of the multiple solutions of the Newton method. I need to assume that s in front of the calculations to define the bulb's outside.
- Compute the Newton for t //is this a single iteration of the root-finding method or do you iterate the method until it becomes really close to the root
It's one step.
i.e. t(n+1) := ((p-1)/p)*t(n) - c*t(n)^(1-p)/p  (p is the power, and c the same c as in the normal bulb. That means I search solutions for t^p+c=0)
- And is this the fractal formula or the actual distance field?
In MB3D I can't change the DE calculation. With JIT I can (as far as I know) only calculate the actual formula (i.e. the calculation of t). It's very simple.
If the formula in the .m3f looks quite complex then partially because I need to work around the JIT/MB3D limitations

M3D .m3f formulas are human readable if they are written in JIT code (as mine are).
Please have a look to the text after the [END] tag - there you can find detail explanations.
Place the .m3f files into the [MB3Root]\M3Formulas directory before you start the program.

#### gannjondal

• Fractal Freshman
• Posts: 5
« Reply #8 on: Yesterday at 07:57:32 PM »

Reason:  In the original versions I used an old, slower version of the calculations.

Changes in JIT_gnj_RealPowNewton_04.m3f:
- Optimized the power calculations. In the new version the calculation of the picture Newton-1a is now about 30 % faster
- Introduced the variable Fake_Bailout -
Just another hack to have the formula working more smooth with MB3D, specifically if using the formula in hybrids.
Setting of the correct 'R bailout' value on MB3D's formula tab is a bit tricky for this for this formula.
If it is too large you will receive plain blocks of inside. If you want/need to increase then increase Fake_Bailout for the same factor (defaults: Fake_Bailout=1 with 'R bailout'=4).
You may set it e.g. to 4 if making hybrids with Mandelbulb type of formulas (with a standard R bailout of 16).
Side effect:   Large pairs of Fake_Bailout and 'R bailout' can improve the performance again. For instance with Fake_Bailout=400 and 'R bailout'=1600 the picture Newton-1a was even 60 % faster than in the previous formula (colors change a bit tough)!
« Last Edit: Yesterday at 09:53:58 PM by gannjondal »

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