A few Mandelbrot variations I discovered (Stay tuned for more!)

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Offline greentexas

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« on: September 06, 2017, 05:17:01 PM »
This is a set of ten Mandelbrot variations I discovered. Some of them may have already been discovered, some of them have no Minibrots, but stay tuned for more fractals.

VARIATION 1
ozr = zr;
zr = zr*zr - zi*abs(zi) + JuliaR;
zi = 2*ozr*zi + JuliaI;

VARIATION 2
ozr = zr;
zr = zr*zr - zi*zi + JuliaR;
zi = 2*ozr*zi*zi + JuliaI;

VARIATION 3
if (zr > 0) {
ozr = zr;
zr = zr*zr - zi*zi + JuliaR;
zi = 2*ozr*abs(zi) + JuliaI;
} else {
ozr = zr;
zr = zr*zr - zi*zi + JuliaR;
zi = 2*abs(ozr)*zi + JuliaI;
}

VARIATION 4
ozr = zr;
zr = zr*zr - zi*zi - 2*zr*zi + JuliaR;
zi = 2*ozr*zi - 2*zr*zi*zi + JuliaI;

VARIATION 5
ozr = zr;
zr = zr*zr*zr - 1.5*zi*zi*zi + JuliaR;
zi = -3*ozr*ozr*zi + JuliaI;

VARIATION 6
ozr = zr;
zr = (zr*zr - zi*zi)*(zr*zr - zi*zi) + JuliaR;
zi = 4*ozr*ozr*zi*zi + JuliaI;


VARIATION 7

ozr = zr;
zr = (zr*zr - zi*zi)*(zr - zi) + JuliaR;
zi = 3*ozr*zi + JuliaI;

VARIATION 8

if (zi > 0) {
ozr = zr;
zr = abs(zr*zr - zi*zi) + JuliaR;
zi = 2*ozr*zi + JuliaI;
} else {
ozr = zr;
zr = (zr*zr - zi*zi) + JuliaR;
zi = 2*ozr*zi + abs(JuliaI);   
}

VARIATION 9

ozr = zr;
zr = (zr*zr - zi*zi) + JuliaR;
zi = -2*abs(ozr)*zi + JuliaI;
ozr = zr;
zr = (zr*zr - zi*zi);
zi = 2*abs(ozr)*zi + JuliaI;

VARIATION 10

ozr = zr;
zr = (zr*zr - zi*zi) + JuliaR;
zi = 2*abs(ozr)*zi + JuliaI;
ozr = zr;

zr = (zr*zr - zi*zi) + JuliaR;
zi = 2*(ozr)*zi + JuliaI;

Offline RedshiftRider

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« Reply #1 on: September 06, 2017, 09:28:07 PM »
Hey Greentexas, good to see you found your way here. Welcome!


Great idea I am curious how they look, could you provide a visual representation of the the different variants?

Offline Softology

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« Reply #2 on: September 07, 2017, 12:19:01 AM »
Assuming I didn't make any mistakes with the formulas, here are some sample images and a zoomed in preview for each of the first 9.

Variation 1





Variation 2





Variation 3





Variation 4





Variation 5





Variation 6





Variation 7





Variation 8





Variation 9





Variation 9 is the Heart Mandelbrot from stardust4ever's variations



Jason.

Offline greentexas

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« Reply #3 on: September 07, 2017, 04:16:41 AM »
Thank you so much! I wouldn't think anyone would be so kind to do that for me! When I created formula #9, it did not look like the Heart. I tried to make sure it wasn't like his 4th Quasi Heart either. Stardust4ever's fractals were an inspiration for me to make this.

Offline RedshiftRider

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« Reply #4 on: September 07, 2017, 04:00:30 PM »
These look quite interesting.


I think I have seen variation 3 before. Isn't that some sort of combination (well all of them probably are) of the m-set and the burning ship?

Offline greentexas

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« Reply #5 on: September 09, 2017, 03:16:48 AM »
Variation 3 is not a Mandelbrot/Burning Ship hybrid like stardust4ever's fractals, but 50% of the numbers used are Mandelbrot/Mandelbar, and the other 50% is Burning Ship.

Let's say we have this psuedocode:
if zr > 0 {
do(Mandelbrot)
} else {
do(Mandelbar)
}

It creates the Perpendicular Mandelbrot.

This formula's psuedocode would be:
if zr > 0 {
do(Perpendicular Burning Ship)
} else {
do(Perpendicular Mandelbrot)
}

making it like a hybrid of a hybrid. There is one Mandelbar, one Mandelbrot, and two Burning Ships.

Offline greentexas

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« Reply #6 on: September 09, 2017, 03:37:45 AM »
Ten more variations! (The first time I tried plotting the Cubic Mandelbrot using Scratch a couple years ago, I think version 15 was a mistake I made. I was pretty silly back then, to be using ABS.)

Variation 20 is a 6th order variation of variation 1.

I'll send an image soon of the ten variations.

VARIATION 11
ozr = zr;
zr = zr*zr*(1 + zi) - (zi*zi * (1 + zr)) + JuliaR;
zi = -2*ozr*zi + JuliaI;

VARIATION 12
ozr = zr;
zr = -(zr*zr - 3*zi*zi)*(zr) + JuliaR;
zi = -abs(3*ozr*ozr - zi*zi)*(zi) + JuliaI;

VARIATION 13
ozr = zr;
zr = (zr*zr*zr - 3*zi*zi)*(zr) + JuliaR;
zi = (3*ozr*ozr*zi - zi*zi)*(zi) + JuliaI;

VARIATION 14
ozr = zr;
zr = (zr*zr - 3*zr*zi*zi) + JuliaR;
zi = (3*ozr*ozr - zi*zi)*(zi) + JuliaI;

VARIATION 15
ozr = zr;
zr = (zr*zr*zr - 3*zr*zi*zi) + JuliaR;
zi = (3*abs(ozr*ozr)*zi - abs(zi)*zi*zi) + JuliaI;

VARIATION 16
ozr = zr;
zr = (zr*zr*zr - 3*abs(zr)*zi*zi) + JuliaR;
zi = (3*abs(ozr*ozr)*zi - (zi)*zi*zi) + JuliaI;

VARIATION 17
ozr = zr;
zr = abs(zr*zr*abs(zr)- 3*abs(zr)*zi*abs(zi)) + JuliaR;
zi = abs(3*abs(ozr*ozr)*zi - abs(zi)*zi*zi) + JuliaI;

VARIATION 18
ozr = zr;
zr = (zr*zr*(zr)-(zi*zr) + 3*(zr)*zi*(-zi)) + JuliaR;
zi = (3*(ozr*ozr*ozr)*zi - (zi)*zi*zi) + JuliaI;

VARIATION 19
ozr = zr;
zr = 2*zr*zi + JuliaI;
zi = ozr*ozr*(ozr + zi) - zi*zi + JuliaR;

VARIATION 20
ozr = zr;
zr = zr*zr - zi*zi*zi;
zi = 2*ozr*zi;
ozr = zr;
zr = zr*zr - zi*zi*zi + JuliaR;
zi = 2*ozr*zi + JuliaI;
« Last Edit: September 11, 2017, 11:11:40 PM by greentexas »

Offline Softology

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« Reply #7 on: September 14, 2017, 12:03:26 AM »
OK, I gave those ones a go too.  A lot of them are distorted stretched "whipped cream" results that are not too interesting, but a few are OK.

Variation 11





Variation 12





Variation 13





Variation 14





Variation 15





Variation 16





Variation 17





Variation 18





Variation 19





Variation 20





All of those were rendered in http://softology.com.au/voc.htm.

Jason.

Offline greentexas

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« Reply #8 on: October 03, 2017, 11:11:54 PM »
Here are ten more variations. The last ten weren't very good, I'll admit it. I discovered variation 29 when I was about nine, and nicknamed it the "sitting cheese curl".

VARIATION 21
ozx = zx;
zx = zx*zx - zy*zy*(3 + 4*zy) + cx;
zy = 2*ozx*zy + cy;

VARIATION 22
ozx = zx;
zx = abs(zx - 0.125*zy);
zy = zy + 0.0625*zx*zx;
zx = zx;
zx = zx*zx - zy*zy + cx;
zy = 2*ozx*zy + cy;

VARIATION 23
ozx = zx;
zx = zx*abs(zx) - zy*zy;
zy = 2*ozx*(zy);
ozx = zx;
zx = zx*zx - zy*zy + cx;
zy = 2*ozx*abs(zy) + cy;

VARIATION 24
ozx = zx;
zx = zx*zx + zy*zy - zy*zy*zy + cx;
zy = -2*ozx*zy + (cy);


VARIATION 25

ozx = zx;
zx = 3*zx*abs(zx)*zy - zy*zy*zy + cx;
zy = -abs(ozx)*ozx*ozx - 3*ozx*abs(zy)*zy + cy;


VARIATION 26

ozx = zx;
zx = zx*zx - zy*zy + cx;
zy = 2*ozx*zy + (cy);
if (zx > zx*sqrt(sqrt(abs(zx)))) {
zy = -zy;  
}


VARIATION 27

ozx = zx;
zx = sqrt(zx*zx + zy*zy)*(zx*zx - zy*zy) + cx;
zy = (-2*ozx*zy)*sqrt(abs(ozx*zy)) - cy;


VARIATION 28

ozx = zx;
zx = (zx*zx - zy*zy) - cx;
zy = (2*ozx*zy) - cy;
ozx = zx;
zx = (zx*zx - zy*zy) - cx;
zy = (2*ozx*zy) + cy;
ozx = zx;
zx = (zx*zx - zy*zy) + cx;
zy = (2*ozx*zy) - cy;
ozx = zx;
zx = (zx*zx - zy*zy) + cx;
zy = (2*ozx*zy) + cy;


VARIATION 29

ozx = zx;
zx = (zx*zx - zy*zy) + zy + cx;
zy = (2*ozx*zy) - zy - cy;


VARIATION 30

ozx = zx;
zx = (zx*zx - zy*zy) + cx;
zy = (2*ozx*zy) + cy;
if (zx - zy * zx > zy) {
zx = zx-zy*zy;   
}

Offline Fraktalist

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« Reply #9 on: October 04, 2017, 12:11:23 AM »
very interesting!

there was this fascinating video over at ff1.com, where these variations were shown as a complete morph-cycle, making it visually obvious how the burning ship (if I remember  corectly) was just a variation of the mset..

I couldn'r find it - if anyone does remember the video, please post here!

Offline Softology

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« Reply #10 on: October 04, 2017, 04:36:55 AM »
Here are ten more variations. The last ten weren't very good, I'll admit it. I discovered variation 29 when I was about nine, and nicknamed it the "sitting cheese curl".

It would be really helpful if you included your own sample pics with formulas.  At least then I know if I have the formula right when I convert them into GLSL for my program.

Offline F. Bernkastel

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« Reply #11 on: December 15, 2017, 10:17:25 AM »
there was this fascinating video over at ff1.com, where these variations were shown as a complete morph-cycle, making it visually obvious how the burning ship (if I remember  corectly) was just a variation of the mset..

Do you mean something like this?
\( z_{n+1} = \left\{\begin{matrix}
\|z_{n}\| > \psi,   & z_{n}^{2} + \textrm{C}\\
\|z_{n}\| \leq  \psi,   & ( |\Re{(z_{n}})| + i\,|\Im{(z_{n}})|)^{2} + \textrm{C}
\end{matrix}\right. \)

Where Ψ is a random value with normal distribution. from 0 to 2.
« Last Edit: February 21, 2018, 08:15:15 AM by F. Bernkastel »
...Then clasp me round the neck once more, and give me one more kiss;
And now mine own dear little girl, there is no way but this.


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