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 Author Topic: 3d Burning Ship & extension to 3d Mset for n=2,6,10..., odd n; & another formul  (Read 8982 times) Description: Sign assignment to Burning Ship fractal creates Mandelbrot Set (read wikis) 0 Members and 1 Guest are viewing this topic.
M Benesi
Fractal Schemer

Posts: 1075

 « on: October 23, 2010, 06:50:13 PM »

I can't believe how obvious the formula is after I discovered it....

There are 2 main varieties, although I prefer the one I'm posting images of.  I stumbled across this formula while playing around with a new fractal type, and decided to apply what I discovered to my old complex compound formula.

You can redo the code in trig form if you want, using dual-complex numbers works about 2 times as fast on my comp...  I'd like to add that my other formula, which combines the 2 3d Mandelbrot varieties, produces far more interesting fractals (at least for higher n z^n).  I'll post it beneath the images...

Code:
r1=sqrt(sqr(sy)+sqr(sz));   // you can do x and y values here instead and generate a different fractal
//  gotta make sure you change the rest of the formula to match if you decide to do so
//  I prefer the way this looks, for whatever reason... anyways-
victor=complex(sx,r1)^n;
bravo=complex(sy,sz)^n;

r3=part_i(victor)*r1^-n;

nx=part_r(victor);
ny=-abs(r3*part_r(bravo));
nz=-abs(r3*part_i(bravo));

//Then you add in your regular x pixel component and the absolute value of your y and z pixel components:

sx=nx+       (pixelr);
sy=ny+  abs  (pixeli);
sz=nz+  abs  (pixelj);    //these values are the starting values of the next iteration...

bailout= abs(sx)+abs(sy)+abs(sz)  // or square 'em if it makes you happy... doesn't make a difference to me

side:top:
rear:front:

This formula is produces way cooler fractals.  While it skews away from the Mandelbrot type a bit, it has more variety... it's just more interesting.

Code:
victor=complex(sx,sqrt(sqr(sy)+sqr(sz)));   //the major difference in this formula is that it rotates sx
bravo=complex(sqrt(sqr(sx)+sqr(sy)),sz);   // vs  sy and sz, but then calculates the sy and sz values
cramden=complex(sx,sy);                        // using the other base mandelbrot formula... Makes an AWESOME fractal
//  you can also switch which part you do the stuff with if you so desire...
r1=cabs(cramden)^-n;
victor=victor^n;
bravo=bravo^n;
cramden=cramden^n;

if (r2mode) {                                      //It's neater when you exchange the y and z parts, however I put this switch in
nx=part_r(victor);                       //  so I could do it the other way as well
nz=-abs(part_i(bravo));
ny=-abs(part_r(bravo)*part_i(cramden))*r1;
} else {
nx=part_r(victor);             //this is the more interesting variety, the default...
ny=-abs(part_i(bravo));
nz=-abs(part_r(bravo)*part_i(cramden))*r1;
}
 « Last Edit: October 29, 2010, 08:38:01 PM by M Benesi » Logged

Paolo Bonzini
Guest
 « Reply #1 on: October 23, 2010, 11:33:06 PM »

nx=part_r(victor);
ny=-abs(r3*part_r(bravo));
nz=-abs(r3*part_i(bravo));

sx=nx+       (pixelr);
sy=ny+  abs  (pixeli);
sz=nz+  abs  (pixelj);    //these values are the starting values of the next iteration...

Why the abs and (for ny and nz) the negation?  The formula without them is, if I did my math right,

Code:
r1=sqrt(y^2+z^2)
sx = x^2-y^2-z^2 + pixelr
sy = 2*x*(y^2-z^2)/r1 + pixeli
sz = 4*x*y*z/r1 + pixelj

which embeds the 2d mandelbrot.
 Logged
M Benesi
Fractal Schemer

Posts: 1075

 « Reply #2 on: October 24, 2010, 12:13:40 AM »

Quote
Why the abs and (for ny and nz) the negation?  The formula without them is, if I did my math right,
It does look like you get a standard 2d cross section if you don't do the abs/negations, however there is something I've got to look into further before I say anything else (perhaps in a couple hours I'll follow through, have to do a few things now); ooohhh I remembered:

You are NOT going to get a 2d Mandelbrot cross section with x + i sqrt(y^2+z^2), because you are always taking the absolute value of the y component :sqrt(y^2) = |y| ....  You could set the value to y's sign though but then you get huge missing chunks out of your fractal (I've tried it with: sign of y, sign of z, sign of y+z... all are discontinuous: huge slices cut right out of the fractal).

Come to think of it, I may not be justified in calling this a 3d Mandelbrot, as it's more along the lines of a 3d Burning Ship fractal.

The best looking 3d rotation based fractal to date is the "beautiful fractal" which is the formula I posted at the bottom of the first post in this thread.  It's got tremendous variety for all z^n... and I extended it to 4d... totallllly amazing.
 « Last Edit: October 24, 2010, 02:25:20 AM by M Benesi » Logged

M Benesi
Fractal Schemer

Posts: 1075

 « Reply #3 on: October 24, 2010, 07:06:05 PM »

I did go ahead and set up the fractal with

$r1=\sqrt{| y*|y|+|z|*z |}$    (if you can tell, I take the absolute value of all the variables so I don't get an imaginary root... but what the heck, ehh?  maybe I should make it imaginary...  if it was imaginary... hrmm interesting...)

if (y+z<0)  then r1=-r1..  which doesn't reduce to a 2d Mandelbrot (while doing something similar with sqrt(y^2+z^2) gives you a discontinuous fractal).

Still need the abs/negation to get the nicest fractals, any which way you do it, although the following method works:

Code:
r1=sqrt(sqr(sy)+sqr(sz));

if (sy>sz) {
z1=complex(sx,r1)^n;
r3=r1^-n;
} else {
z1=complex(sx,-r1)^n;
r3=(-r1)^-n;
}
if (sy*sz<0) {
z3=complex(sx,-r1)^n;
r4=(-r1)^-n;
} else {
z3=complex(sx,r1)^n;
r4=r1^-n;
}
z2=complex(sy,sz)^n;

nx=part_r(z1);
ny=part_i(z1)*part_r(z2)*r3;
nz=part_i(z3)*part_i(z2)*r4;

Then add in pixel components.  It's still nicer if you abs/negate it however.
 « Last Edit: October 25, 2010, 12:02:19 AM by M Benesi » Logged

Jesse
Fractal Schemer

Posts: 1013

 « Reply #4 on: October 28, 2010, 06:59:08 PM »

The formula in the first post produces one of the most interesting power 2 bulbs i have seen!
It has similarities with Msltoes symmetric variations.

Without these absolute (foldings?) it is a cosine bulb, that is what i discovered.. but maybe i did something different like always

It seems that the search for a holy grail has become the direction of combining the box and the bulb somehow,
the "boxers" are adding rotations and the "bulbers" more foldings

A detail of the power 2 bulb:

<img src="http://www.fractalforums.com/gallery/3/1127_28_10_10_6_46_52.jpeg" />

PS: i attached the power 2 formula for M3D if someone is interested, hope you dont mind.
M Benesi
Fractal Schemer

Posts: 1075

 « Reply #5 on: October 29, 2010, 05:38:34 AM »

Thanks Jesse!

The first formula in the first post is the 3d variety of the Burning Ship fractal (without the -y component as I set y to all positive in the equation).  The Burning Ship fractal is simply a 2d Mandelbrot with that uses the absolute value of the real and imaginary components each iteration... ...  it's like a Mandelbrot without +/-.  As the formula produces an EXACT replica of the burning ship, this tells us that the only thing we need to do is assign signs correctly, like I did in that other thread, to make a perfect 3d z^2 Mandelbrot with no singularities (it's in the "singularity" thread in this subforum).

The second formula (first post) though... now that is fricken awesome.

Some buildings in the z^4 version:

It is just awesome....

Although now that we have a 3d Mandelbrot that works for z^2,6,10... and all odd n, I found us some seahorses (z^6 though, should search the same location in z^2, as that is where they might be):

 « Last Edit: October 29, 2010, 05:43:59 AM by M Benesi » Logged

M Benesi
Fractal Schemer

Posts: 1075

 « Reply #6 on: October 29, 2010, 06:19:42 AM »

But none of those is worthy of the true power of the 2nd formula.  The face of Anachranox (4d) is:

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Jesse
Fractal Schemer

Posts: 1013

 « Reply #7 on: October 29, 2010, 04:30:26 PM »

2 questions about the 2nd formula, that starts with:
Code:
victor=complex(sx,sqrt(sqr(sy)+sqr(sz)));   //the major difference in this formula is that it rotates sx
bravo=complex(sqrt(sqr(sx)+sqr(sy)),sz);   // vs  sy and sz, but then calculates the sy and sz values
cramden=complex(sx,sy);                        // using the other base mandelbrot formula... Makes an AWESOME fractal
//  you can also switch which part you do the stuff with if you so desire...
r1=cabs(cramden)^-n;

What computes the cabs function?
I took in my first attempts the realpart of bravo, seems to make sense...

And the pixel addition is like in the first formula with abs on y and z?

This produces some weird and wired stuff, has to explore more until i could say what formula i like more.

Two images of the second formula, first without changing y and z, and the second with changing y and z
(and with my assumptions about the 2nd formula):

 BEnesiN2p2.jpg (243.62 KB, 800x600 - viewed 544 times.)  BEnesiN3p2.jpg (244.34 KB, 800x600 - viewed 353 times.) Logged
M Benesi
Fractal Schemer

Posts: 1075

 « Reply #8 on: October 29, 2010, 08:33:21 PM »

Quote
What computes the cabs function?
oh, sheesh, didn't even think of explaining that portion...

z = complex (x,y)    creates a complex number z= x + i y

cabs (z)   computes the absolute value (magnitude) of a complex or quaternion number in ChaosPro.

So if z= x+ i y        cabs(z) = $\sqrt {x^2 + y^2}$

Quote
And the pixel addition is like in the first formula with abs on y and z?
Yes.     Although I am thinking about trying the sign assignment method that works for the first formula (making the Burning Ship into z^2 Mandelbrots) and seeing what it does with the second formula.

I like the switched y and z component formula better than the "normal" method as well (it produces interesting patterns).
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Jesse
Fractal Schemer

Posts: 1013

 « Reply #9 on: October 29, 2010, 10:20:55 PM »

Thank you, it is nearly selfexplaining but i wanted to be sure before i make a formula.
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visual.bermarte
Fractal Fertilizer

Posts: 355

 « Reply #10 on: October 30, 2010, 05:17:51 PM »

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Jesse
Fractal Schemer

Posts: 1013

 « Reply #11 on: October 30, 2010, 06:27:47 PM »

Amazing, is this a julia animation from the first formula?

It shows nice attributes of this formula, i still dont know if i like the second one more or not...

nevertheless i attached the power 2 version of the second one with changed z and y.
bib
Global Moderator
Fractal Senior

Posts: 2070

At the borders...

 « Reply #12 on: October 30, 2010, 06:44:29 PM »

Jesse, I just noticed you've attached some new formulae here. Are there any others I might have missed recently??
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Between order and disorder reigns a delicious moment. (Paul Valéry)
Jesse
Fractal Schemer

Posts: 1013

 « Reply #13 on: October 30, 2010, 06:48:17 PM »

Jesse, I just noticed you've attached some new formulae here. Are there any others I might have missed recently??

Nope
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bib
Global Moderator
Fractal Senior

Posts: 2070

At the borders...

 « Reply #14 on: October 30, 2010, 06:57:00 PM »

yes. new M3D is really fast.Thanks