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Author Topic: (Other) (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid  (Read 1358 times)

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Offline claude

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(Other) (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« on: January 30, 2018, 12:19:42 PM »
Formula suggested in the gallery by realflow100: One iteration of Mandelbrot, then one of Burning Ship, then two more Mandelbrot, repeat.  Has Tricorn (Mandelbar) and Mandelbrot mini-sets.

306 keyframes rendered with 'et', using Newton-Raphson zooming for navigation and perturbation techniques for rendering.  Coloured with atom domains and exterior distance estimation (using derivatives). Keyframes assembled into video using the mightmandel/extra/zoom program.  Rudimentary soundtrack made with Pure-data, controlled by video analysis using ffmpeg.

https://mathr.co.uk/misc/2018-01-30_mandelbrot_burningship_mandelbrot_mandelbrot_hybrid.mp4 (27MB)

Resolution 320x180, 75mins render time (using quad core AMD Athlon(tm) II X4 640 Processor at 3GHz), 3mins run time.


Linkback: https://fractalforums.org/index.php?topic=781.0

Offline claude

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Offline greentexas

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #2 on: January 31, 2018, 10:26:35 PM »
Well done!

That was an unusual and unique fractal. I rarely consider iterating two formulas one after the other. Another thing that caught my attention is that you used the slashes in a way I was not familiar with. Normally, when I mention formulas, I think of stardustforever's BS/Buffalo hybrid fractal that has a real Burning Ship component and an imaginary Buffalo.

I posted a picture of a 3rd order variant to this fractal.

Offline greentexas

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #3 on: January 31, 2018, 10:36:16 PM »
That fractal was creative and is worthy to be an idea for future fractal formulas that I come up with :). I explored around with that fractal a little bit. I have a fractal-themed e-mail chain and will mention this fractal in tomorrow's e-mail.

I did 4th and 5th order variations also for you.

Offline Softology

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #4 on: February 01, 2018, 01:10:45 AM »
I must be doing something wrong as I get this for the power 2 variant...



This is my inner iterations loop

Code: [Select]
for(i=0; i<maxiters; i++)
{
double x;
double y;
if (mod(i,4)==0) {
//Mandelbrot
x = (z.x * z.x - z.y * z.y) + c.x;
y = (z.y * z.x + z.x * z.y) + c.y;
}
if (mod(i,4)==1) {
//Burning Ship
x = z.x * z.x - z.y * z.y -c.x;
y = -2.0 * abs(z.y * z.x) - c.y;
}
if (mod(i,4)==2) {
//Mandelbrot
x = (z.x * z.x - z.y * z.y) + c.x;
y = (z.y * z.x + z.x * z.y) + c.y;
}
if (mod(i,4)==3) {
//Mandelbrot
x = (z.x * z.x - z.y * z.y) + c.x;
y = (z.y * z.x + z.x * z.y) + c.y;
}
magnitude=(x * x + y * y);
if(magnitude>bailout_squared) break;
z.x = x;
z.y = y;
}

Can anyone spot an error?

Jason.

Offline claude

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #5 on: February 01, 2018, 01:40:05 AM »
Something to do with conflicting definitions of Burning Ship.  To match my code, you should replace

Quote
Code: [Select]
if (mod(i,4)==1) {
//Burning Ship
x = z.x * z.x - z.y * z.y -c.x;
y = -2.0 * abs(z.y * z.x) - c.y;
}

with

Code: [Select]
if (mod(i,4)==1) {
//Burning Ship
x = z.x * z.x - z.y * z.y + c.x;
y = 2.0 * abs(z.y * z.x) + c.y;
}

and draw with the imaginary axis increasing towards the bottom of the image (real axis increases to the right as normal).

As I understand it, power p Burning Ship is   x+i y -> (|x| + i |y|)^p + c

Offline Softology

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #6 on: February 01, 2018, 02:37:57 AM »
Bingo!  Thanks Claude.

I had the -2 to get the Burning Ship facing the "right way" as in most images of it online.

Jason.

Offline claude

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #7 on: February 01, 2018, 06:34:56 AM »
I posted a picture of a 3rd order variant to this fractal.

With repeat( cubic Mandelbrot -> cubic Burning Ship -> cubic Mandelbrot -> cubic Mandelbrot -> ) I get the attached.  Slightly different from yours, don't know why yet...

Has a variety of mini-sets, though the #4 and #5 below are much rarer than the #2 and #3, which both occur quite often.  Thanks for the idea to use higher powers :)

Offline claude

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #8 on: February 01, 2018, 06:56:33 AM »
The two main different kinds of minis in the cubic variant can occur in close proximity.

Offline Softology

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #9 on: February 01, 2018, 09:06:47 PM »
I noticed the same thing in the power 2 version.


Offline Softology

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #10 on: February 01, 2018, 10:38:16 PM »
I get different results again for powers 3, 4 and 5.  I am assuming that we all use the same mandelbrot code, so the differences must come down to how we calculate the burning ship steps.  For reference here are my results and burning ship codes.

Power 3



Code: [Select]
//Burning Ship
double zisqr = z.y * z.y;
double zrsqr = z.x * z.x;
x = (zrsqr-(zisqr*3.0))*abs(z.x)-c.x;
y = ((zrsqr*3.0)-zisqr)*abs(z.y)+c.y;

Power 4



Code: [Select]
//Burning Ship
double zisqr = z.y * z.y;
double zrsqr = z.x * z.x;
x = zrsqr * zrsqr + zisqr * zisqr - 6.0 * zrsqr * zisqr -c.x;
y = 4.0 * abs(z.x * z.y) * (zrsqr - zisqr) +c.y;

Power 5



Code: [Select]
//Burning Ship
double zisqr = z.y * z.y;
double zrsqr = z.x * z.x;
double zisqrsqr = zisqr * zisqr;
double zrsqrsqr = zrsqr * zrsqr;
double zrzisqr = zrsqr * zisqr;
x = abs(z.x) * (zrsqrsqr - 10.0 * zrzisqr + 5.0 * zisqrsqr) + c.x;
y = abs(z.y) * (5.0 * zrsqrsqr - 10.0 * zrzisqr + zisqrsqr) + c.y;

From memory, when I implemented the Burning Ship formulas I used stardust4ever's reference images, ie









Jason.

Offline greentexas

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #11 on: February 02, 2018, 12:21:33 AM »
Here's why the confusion happened (in part):

I used

abs(zr*zr - zi*zi + juliaR)
abs(2*zr*zi + juliaI)

for the Burning Ship, even though I knew better. My formula was wrong! :fp:

Now that I used stardustforever's formulas, I did some more accurate drawings for orders 3 through 5.

Offline Softology

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #12 on: February 02, 2018, 12:53:33 AM »
OK, with some tweaks (I had a few changed signs from stardust4ever's versions) I now get the same results as you do.  Here are powers 2 to 5 with the Burning Ship code.

Power 2



Code: [Select]
//Burning Ship
x = z.x * z.x - z.y * z.y + c.x;
y = 2.0 * abs(z.y * z.x) + c.y;

Power 3



Code: [Select]
//Burning Ship
double zisqr = z.y * z.y;
double zrsqr = z.x * z.x;
x = (zrsqr-(zisqr*3.0))*abs(z.x)+c.x;
y = ((zrsqr*3.0)-zisqr)*abs(z.y)+c.y;

Power 4



Code: [Select]
//Burning Ship
double zisqr = z.y * z.y;
double zrsqr = z.x * z.x;
x = zrsqr * zrsqr + zisqr * zisqr - 6.0 * zrsqr * zisqr +c.x;
y = 4.0 * abs(z.x * z.y) * (zrsqr - zisqr) +c.y;

Power 5



Code: [Select]
//Burning Ship
double zisqr = z.y * z.y;
double zrsqr = z.x * z.x;
double zisqrsqr = zisqr * zisqr;
double zrsqrsqr = zrsqr * zrsqr;
double zrzisqr = zrsqr * zisqr;
x = abs(z.x) * (zrsqrsqr - 10.0 * zrzisqr + 5.0 * zisqrsqr) + c.x;
y = abs(z.y) * (5.0 * zrsqrsqr - 10.0 * zrzisqr + zisqrsqr) + c.y;

Do we all have the "correct" matching results now?

Jason.

Offline claude

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #13 on: February 02, 2018, 03:44:30 PM »
Do we all have the "correct" matching results now?

I only did power 2 and 3, but they look like they match mine.

Offline Softology

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Re: (Mandelbrot / Burning Ship / Mandelbrot / Mandelbrot) Hybrid
« Reply #14 on: February 03, 2018, 09:38:28 PM »
Here is a quick 4K 60fps video of some power 2 zooms.  Some nice areas in these.


Jason.


xx
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