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Reading again the triplex algebra threads (http://www.fractalforums.com/index.php?topic=2062), I have been thinking of a strange idea.
What happens if you do a two stage power 8 bulb, doing two different power formulas?
For example, you can do the following;

1) Power 4 Sine formula, without adding C;
2) Power 2 Cosine Bulb, adding C. (Total power; 2 x 4 = 8)

It has interesting features and it looks different from the original bulbs.

Another possible thing is to do a p4 cosine and p2 sine, or even a power two, then a power four mixed. 88) :yes: "Pretty" images coming soon :alien: :gum:

Closeup of the whole bulb; has cool details and interesting features :D

(http://orig12.deviantart.net/7748/f/2017/058/9/d/bulb_mixed_by_dark_beam-db0kppm.png)

Image of a spinal julia set close to (0.6;0.3;-0.3) in inside mode

(http://img09.deviantart.net/3aad/i/2017/058/c/7/air_elemental_masquerade_by_dark_beam-db0koqt.png)

Closeup of the bulb obtainable doing a power 4 cosine then a sine squaring and adding C:

(http://img06.deviantart.net/98f6/i/2017/058/6/3/bulb67_by_dark_beam-db0krff.png)

Really an amazing combination - and interesting pictures!


But I hope that you know that you are in danger, Luca :hurt: :

Your idea kicked my brain to search for the articles that list all the variants besides sine, and cosine.

So, if you take all 4-2, 2-4, or 2-2-2 combinations of the variants listed at
http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/

or the 512 cos/sin combinations explained at
https://softologyblog.wordpress.com/2011/07/21/new-mandelbulb-variations/
and the related pages

or
http://softology.com.au/gallery/gallerymandelbulb.htm

....

No, I need to stop, I get a headache  :vomit:

Also MB3D would probably not start with soo many formulas, so that's maybe not a good idea.


Although I love mutations....

Hey my friend ;)
I "only" tried to combine the most promising unimplemented bugman formulas mixing them with sine and cosine this leads to four variants each formula total 12.
As for the all other variants - I did not exactly remember all those...
But definitely not all combos are good looking. So ... we need to be a bit choosy
You still can code them into your formulas though. (Using Andreas programmable engine). Remember that you *need* to simplify trigonometric funcs;
Each one of those expressions simplify to rational / way simpler expressions for integer n.
That gives a huge boost to render speed.

Side note.

Just to blast to insanity the number of variations. It is possible to use different n for the two kinds of angles you use.
Example use (4;2) for the first and (2;4) for the second to give a different power 8 bulb, even different from the previously listed mixes!!!  :troll: :horsie: :nono:

Help appreciated for this issue;
Is anyone able to do the following mixing?
1. Calculate pow3 sine bulb
2. Calculate pow5 sine bulb
3. Using a "cosine product" (or whatever) multiply the resulting triplexes.
Note!!!
You should get a continuous formula BUT i get cuts and magnitude is perturbed. Why!!!
4. Add c

@ darkbeam. I don't quite understand :embarrass: I had to google "cross product", I vaguely remember that it was fun at school. :)
You would have to show me, and then I should get the exactly the same cuts  as you LOL

@ gannjondal
There are so many different ways of combining. Bulbs are a mega-infinity on there own.

I have tested mixing them with msltoes Toroidal, and the  mbulb variations of Bermarte and Kali, using just simple iteration_count based combinations.

mclarekin  O0 you did not see the monumental thread Triplex Algebra?

http://www.fractalforums.com/index.php?topic=2062.msg9476#msg9476

Given two triplex you can do a product between them
The product law itself depends on how you define angles.
If you define angles using the Cosine variation the very law changes.
(Also. The azimuthal, reversed ecc. bulbs currently were only tested in sine variants. Using arccos will lead to new variants ...)

@ Darkbeam
I have seen it lots of times, and It always looks far too complex for me. :o

I can actually understand more of it these days, but still not enough. So with bulbs I generally use the time proven method of random trial and error .

You and gannjondal  have managed to side track me away from what I was working on :police: :fiery: but as always it is fun :beer: :beer: :beer:

@DarkBeam: 
Agree, not all of the examples I have seen are beautiful (or even useful), and also some of the variants (like positive vs negative variants) may be too equivalent to show good results. Thus you have a good argument to limit the work  :dink:
You are also right, I should simplify my formulas. Maybe I will check once I find the mood to do so. As of now I like it to have the formulas a bit better readable.
And, ah, I thought optimization is your task  :tongue1:


Note:  I'm writing my answer here as I think that your question here is almost the same as in 'my' thread...

Ok. Maybe I'm already getting stuck in the mud behind the beautiful surface of the Mandelbulb  :dink: , but:
Because I know that I usually make too many errors when calculating polynoms I started to think about your question from the perspective of spheric coordinates.
And from a pure spheric perspective I'm not exactly sure what the described mixture of q³ and q⁵ should introduce on new behavior.
I have read a bit in the big triplex thread, especially:  the spheric definition of a multipication in one of the pictures in the begin, and also the discussion between David Makin, Paolo Bonzini, and Timeroot between Jan-19th and 21th 2010.
As a consequence from all that I would say:  As long as one can use pure spheric numbers (and so far I believe to understand now I can do so as long as I make only multiplications of powers) then q³*q⁵=q⁸ , and nothing does change...

I don't really know whether this is still valid if one always calculates with cartesian coordinates to speed up. I assume that it should come to the same result (and would be a good prove for correct programming  :)) , but I'm not sure. I have tried a bit, but got some mess only right now.

What may make sense would be if one would multiply different bulb types, i.e.
1. Calculate pow n bulb variant 1 (say, sine)
2. Calculate pow (8-n) bulb variant 2 (say, cosine)
3. Multiply the resulting triplexes (by the way:  Did you ever have seen another cartesian triplex multiplication than the sine version that is explained in the triplex math thread?).

@ mclarekin:
Thank you for your own mixture example :)
You are absolutely right, there are already too many bulb versions out in the world now. And starting to mix them may bring us abroad  :alien:

To be honest, I also like all this mixing, and mutating etc too much.
However, maybe we should keep only those results which show something really new/amazing ...

Erm...
You did not pay attention to one important detail ;)
The two basic bulbs are indeed calculated using the same law, but the PRODUCT law is different  :) - so you can do two cosine bulbs then multiplicate each other using a product law of a different theory. Bugman gave the product law for sine but it exists one for cosine too. (I am unsure how to find it exactly).
I already tried but I get a wrong result for unknown reason.
I think the product law fails on different power bulbs in simple words. But it should not!
Every bulb and 3D mandelbrot type has a peculiar associated product law.
And of course; ... no. :D
Really; (q^5) x (q^3) differs from q^8 because Triplex algebra as a whole is not commutative or associative. :) (so also the order is important). Neither is (q^2)^4 = q^8. Bugman confirmed those relations in the early days of Mandelbulb.
This further widens the field of possibilities for bulbs though.

Ouch. Seams that it's now me who stucks in the mud.   :banginghead:
Even if it may be valid to ask about the differences between the multiplication in spheric, and cartesian coordinates - my list makes not even sense when thinking about speric coordinates. Thus... sorry :)
It seams that I should remain quiet (or at least as modest as mclarekin, as a minimum) as long as I am not able to calculate everything by my own. Which will be virtually never  :secret:

But ok. I have tried your suggestion somehow, using the cartesian methods from the big thread (i.e. sine variant for both, powers, and multiplication).
Also after correcting a few errors I also get something with cuts  :sad1: :
(http://orig02.deviantart.net/ca3a/f/2017/065/f/e/pmg_004_v191_5_by_gannjondal-db1er2m.jpg)    (http://orig13.deviantart.net/487e/f/2017/065/f/d/pmg_003_v191_5_by_gannjondal-db1er2v.jpg)

You know the thread better than me, so you may know whether there is a point at which one of the rules gets corrected.
But maybe comparing the errors does help.
Thus here's my variant (you can replace the 1 in x1,tmp1,rho1 etc with 3, and the according 2 with 5 to match your own variant; I wanted to remain more general):

Oh no! It is exactly the result I got. :'(
So there is no way to mix bulbs using triplex multiplication in the normal way.
And I don't know why.
I think the operator only works in special conditions.

Also.

I tried to fix the multiplication using different expressions for r1 and r2.
The results look interesting but nothing fixes the troubles.

And do not worry always share your thoughts friend :beer:

(http://img01.deviantart.net/fcd3/i/2017/070/1/d/you_go_first_by_pupukuusikko-db1ws8z.jpg)

Render by Pupukuusikko - "CosSinTwoStageInv, inside render, Julia from 'spine' region"

http://www.deviantart.com/art/You-go-first-668337155

 :D