Suggestion for a formula

[Kali]

Hi Jesse,

What if you take my formula and add two more combinations of mandelbrots rotated 45 degrees around the vertical axis?

A way for doing this:

I'll name the coordinates x, y and z.

First, for the first pair of orthogonal mbrots:

c1=(z,x)
c2=(z,y)

then for the rotation, a complex value R containing:

R=(x,y)

rotate 45 degrees: R=R*(0.707106781186547, 0.707106781186547)

then get the new x and y:

x=R.realpart
y=R.imaginarypart

the C3 and C4 value with the x,y rotated:

c3=(z,x)
c4=(z,y)

Then, the four iterations for any power:

f1=f1^@power+c1
f2=f2^@power+c2
f3=f3^@power+c3
f4=f4^@power+c4

bailout: f1+f2+f3+f4>32 or whatever you did to get a less noisy version

I did it with my rudimentary 3D plotter and I think the results might be interesting

What do you think?

[DarkBeam]

Hey, (1,1) doesn't rotate by 45°... Use ( 0.707106781186547 , 0.707106781186547 ) 

[Kali]

Ok, sorry... I'm really new at complex numbers 
However, it rotates 45º but scaling up the vector.
Thanks, I edited the post.

[Jesse]

I have not the time for it at the moment.

A little problem here is that all complex numbers must kept somewhere in memory, that is a bit tricky to do with the current record of variables in m3d...

[Kali]

That's ok, just asking. Thanks anyway...

[Kali]

I figured out how to use Traffassel's "Gestaltlupe" to render the formula.

However, I'm having troubles getting some nice renders and close-ups.

I'll ask him for help.

This is what I got so far:

[trafassel]

Some renderings ...

[Kali]

Thanks Trafassel! I think I see the "antennas" of the previous orthogonal formula are now pointing to multiple directions instead of being mostly vertical like before (althought they had angles, there were made of "orthogonal steps")
I'll look the proyect files you pointed in the private message, and see what can I do. I want to explore this object more closely.
Also I'm working on some optimizations to render the formula. One thing is that I'll use a minimum iteration value, that leads to better detail level and less "boxy" shapes... I had great results with mandelbulb 3D by doing this on the previous formula, but it slows down a lot the rendering time.

[DarkBeam]

Can you write this formula in function of x y z w, Cx... to Cw, constants and without any additional variable? Try to simplify the expressions with Mathcad or something
If you report it to plain it could be easy to assemble!

[Kali]

Can you write this formula in function of x y z w, Cx... to Cw, constants and without any additional variable? Try to simplify the expressions with Mathcad or something
If you report it to plain it could be easy to assemble!

DarkBeam, I think there's not a very simplified version of the formula, because the concept is to iterate separated formulas and then testing the sum of them for the escape condition. The idea is very simple thought. Here's my first post about it (and I think the first in the forum after introducing myself)
http://www.fractalforums.com/help-and-support/help-i%27m-looking-for-an-easy-way-to-render-my-formula/

Jesse made the formula for Mandelbulb3D: http://www.fractalforums.com/mandelbulb-3d/a-formula-suggested-by-kali/
Too bad it was not included in the latest version

The new formula is similar but the combination is of 4 2D-Mandys at 45 degrees.
I'll try to explain it better for you, it's pretty simple:

Instead of using only x,y,z points for the formula, I'll use xa,ya,xb,yb and the vertical axis z (leave w out of here). This means I use two x-y planes, and they are rotated 45 degrees around the vertical axis z. So if we thing of the x,y axis describing a cross (+), the xa,ya,xb,yb combination describes something like an asterisk ('+' plus 'x'). Along the vertical z axis, 4 regular 2D Mandelbrots are calculated by separated iterations, sharing z as the real part, and the imaginary parts are xa,ya,xb,yb respectively for each mandy out of 4. If you get this, you will understand the formula:

Let's name the function of each Mandy k1,k2,k3,k4, and the constants c1,c2,c3,c4 (all complex variables)

For each point, define the constants this way:
c1=z+xai
c2=z+yai
c3=z+xbi
c4=z+ybi

Then iterate:
f(k1)=k1*k1+c1
f(k2)=k2*k2+c2
f(k3)=k3*k3+c3
f(k4)=k4*k4+c4

and test if the sum of all their modules is greater than bailout value:
|k1|+|k2|+|k3|+|k4|>128

That's all, hope you understand it, if not tell me so!

[Jesse]

Jesse made the formula for Mandelbulb3D: http://www.fractalforums.com/mandelbulb-3d/a-formula-suggested-by-kali/
Too bad it was not included in the latest version

oh sorry, wasnt my intent.  just forgot it 

[DarkBeam]

Instead of using only x,y,z points for the formula, I'll use xa,ya,xb,yb and the vertical axis z (leave w out of here). This means I use two x-y planes, and they are rotated 45 degrees around the vertical axis z. So if we thing of the x,y axis describing a cross (+), the xa,ya,xb,yb combination describes something like an asterisk ('+' plus 'x'). Along the vertical z axis, 4 regular 2D Mandelbrots are calculated by separated iterations, sharing z as the real part, and the imaginary parts are xa,ya,xb,yb respectively for each mandy out of 4. If you get this, you will understand the formula:

Let's name the function of each Mandy k1,k2,k3,k4, and the constants c1,c2,c3,c4 (all complex variables)

For each point, define the constants this way:
c1=z+xai
c2=z+yai
c3=z+xbi
c4=z+ybi

Then iterate:
f(k1)=k1*k1+c1
f(k2)=k2*k2+c2
f(k3)=k3*k3+c3
f(k4)=k4*k4+c4

and test if the sum of all their modules is greater than bailout value:
|k1|+|k2|+|k3|+|k4|>128

That's all, hope you understand it, if not tell me so!

But final xyz is f(k1), f(k2) ... or sum?

[Kali]

Jesse made the formula for Mandelbulb3D: http://www.fractalforums.com/mandelbulb-3d/a-formula-suggested-by-kali/
Too bad it was not included in the latest version

oh sorry, wasnt my intent.  just forgot it 

Don't worry, Jesse, I was just being a little over dramatic 
Also don't feel pushed to include it, it's YOUR software after all!

But final xyz is f(k1), f(k2) ... or sum?

Uh? mmm... sum? 

[Jesse]

Don't worry, Jesse, I was just being a little over dramatic 
Also don't feel pushed to include it, it's YOUR software after all!

To late, its already in there 

Btw, it would be no big prob to share the project with others who want to develop... but its Delphi and not really easy to understand what i have done...

But final xyz is f(k1), f(k2) ... or sum?

Uh? mmm... sum? 

What i have done in the smooth version is adding the vecs + averaging the veclengths and scale the sum so it matches this length.
A fast but more inaccurate way compared to slerp.
(of course still better than just adding)