• January 16, 2021, 09:42:16 PM

### Author Topic:  How many different ways are there to show such set ?  (Read 1349 times)

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• Fractal Frogurt
• Posts: 434
##### How many different ways are there to show such set ?
« on: November 21, 2020, 07:20:59 PM »
Contest !!!!

Here are 22 different images of imploded cauliflower Julia set (edit :  here c = 0.35): https://gitlab.com/adammajewski/implodedcauliflower and the source code for creating them ( c/bash)

Can you add new method/algoritm ?
Can make it  better ? ( in any way )

Below one example ( BD/J inverted plane )

« Last Edit: November 22, 2020, 06:53:03 AM by Adam Majewski, Reason: c »

#### Caleidoscope

• Moderator
• Posts: 742
##### Re: How many different ways are there to show such set ?
« Reply #1 on: November 21, 2020, 11:56:54 PM »
This would have been a nice idea for the 'technical' challenge

#### gerrit

• 3f
• Posts: 2288
##### Re: How many different ways are there to show such set ?
« Reply #2 on: November 22, 2020, 02:01:21 AM »
Orbit trap using ratio of smallest and next-to smallest |z|.

#### xenodreambuie

• Fractal Friar
• Posts: 105
##### Re: How many different ways are there to show such set ?
« Reply #3 on: November 22, 2020, 05:04:20 AM »
With a branching texture and vortex transform.

• Fractal Frogurt
• Posts: 434
##### Re: How many different ways are there to show such set ?
« Reply #4 on: November 22, 2020, 06:51:32 AM »
With a branching texture and vortex transform.

Can you describe more ?
What program do you use ?
What is a branching texture ? ...

It looks similar to
http://www.juliasets.dk/RatioField-filer/ratiof28.jpg

Do you use similar technique ?
« Last Edit: November 23, 2020, 09:45:48 PM by Adam Majewski, Reason: link »

• Fractal Frogurt
• Posts: 434
##### Re: How many different ways are there to show such set ?
« Reply #5 on: November 22, 2020, 08:29:57 AM »

I'm not sure if it is good image because critical point and it's preimages are visible
compare http://www.math.titech.ac.jp/~kawahira/gallery/movies/0_1.avi
and http://www.math.titech.ac.jp/~kawahira/gallery/movies/0_1_050.png  on page http://www.math.titech.ac.jp/~kawahira/gallery/movies.html

Code: [Select]
/*  The dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:[1] d = a1b1 + a2b2  */double cdot(double complex a, double complex b){   return creal(a)*creal(b)+cimag(a)*cimag(b); }// // output // double GiveReflection(double complex z )  {   int i=0; // iteration    int iMax = 2000;      // https://en.wikipedia.org/wiki/Complex_quadratic_polynomial#First_derivative_with_respect_to_z   double complex dz = 1.0; // derivative with respect to z    double reflection = 0.0; //        double h2 = 1.5 ; // height factor of the incoming light   double angle = 45.0/360.0 ; // incoming direction of light in turns    double complex v = cexp(2.0*angle *M_PI* I); // = exp(1j*angle*2*pi/360)  // unit 2D vector in this direction   // incoming light 3D vector = (v.re,v.im,h2)    // https://en.wikipedia.org/wiki/Lambertian_reflectance      double  complex u;         z  = z*z+c; //    dz = 1.0;      for(i=0;i<iMax;i++)    {        dz = 2.0*dz*z ;      z  = z*z+c; //                         if(cabs(z) > ER_NP)       { // exterior        u = z / dz;        u = u / cabs(u);        reflection =  cdot(u, v) + h2;  // use the simplest model for the shading: Lambert, which consists in using the dot product of (x,y,1) with a constant vector indicating the direction of the light.        reflection = reflection/(1.0 + h2);  // rescale so that t does not get bigger than 1        if (reflection<0.0) reflection =0.0;        break;            }    }       return reflection;   }// Potential to colorunsigned char ComputeColorOfNP(complex double z){//https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/Mandelbrot_set#Normal_map_effect      double reflection;  unsigned char iColor;           // compute    reflection = GiveReflection( z);        //   //if (reflection <  )    //{ /*  interior  */    //  iColor = 0;}  //else // exterior             { iColor = 255 * reflection;}       return iColor;      }

#### xenodreambuie

• Fractal Friar
• Posts: 105
##### Re: How many different ways are there to show such set ?
« Reply #6 on: November 22, 2020, 08:32:13 AM »
Can you describe more ?
What program do you use ?
What is a branching texture ? ...

As field lines in exterior regions branch at each iteration boundary, one way to get a smooth branching pattern is by interpolating between the iterations, as in the triangle inequality average and related coloring methods. But for general procedural textures it is desirable that they can be calculated purely from the coordinates of a point (plus general information such as the degree of the formula.) The most obvious coordinates are continuous potential and field angle. Some patterns can be designed with a shape that branches within a unit square, so that the top edge matches with half of the bottom edge (or ideally, any moderate number of branches so it can work with higher degree formulas.) The pattern I used here is just an abs(sin), with interpolated frequency between 1 and 2 (or the degree, if greater than 2). So if I use a cubic formula it does a three to one branch.

The software is Jux, but a pre-release version for the swirly transformation.

• Fractal Frogurt
• Posts: 434
##### Re: How many different ways are there to show such set ?
« Reply #7 on: November 22, 2020, 10:03:23 AM »
Thx.  Image is cool and Jux is a great  software http://www.xenodream.com/jux.htm

As field lines in exterior regions branch at each iteration boundary ....

What do you mean by field lines? I have found in chm :

Quote
Field Lines are divisions of potential in a direction perpendicular to the potential gradient

Should it be : (external) rays are simply the curves that run perpendicular to the equipotential curves  = along potential gradient ??
Seems similar to  the external rays = field lines of Douady-Hubbard potential
https://en.wikipedia.org/wiki/External_ray

Sho

Here some external rays  branch at critical point and it's preimages
https://commons.wikimedia.org/wiki/File:Dynamical_plane_with_branched_periodic_external_ray_0_for_map_f(z)_=_z*z_+_0.35.png#file
« Last Edit: November 22, 2020, 10:36:52 AM by Adam Majewski »

#### xenodreambuie

• Fractal Friar
• Posts: 105
##### Re: How many different ways are there to show such set ?
« Reply #8 on: November 22, 2020, 10:02:58 PM »

What do you mean by field lines? I have found in chm :
"Field Lines are divisions of potential in a direction perpendicular to the potential gradient"

Should it be : (external) rays are simply the curves that run perpendicular to the equipotential curves  = along potential gradient ??
Seems similar to  the external rays = field lines of Douady-Hubbard potential
https://en.wikipedia.org/wiki/External_ray

Yes, in a potential field the definition of field lines is lines that are perpendicular to equipotential curves. External rays are field lines in an exterior region that are continuous.

But the most convenient way to get a metric perpendicular to equipotentials is to take the arctan2 of the point at bailout. That gives field lines (or lines of longitude) that branch at each iteration boundary in exterior regions. In interior regions they do not branch, so it's easy to apply continuous textures.

• Fractal Frogurt
• Posts: 434
##### Re: How many different ways are there to show such set ?
« Reply #9 on: November 23, 2020, 10:23:06 PM »
Cool. Masterpiece. I like everything. Nice color gradient.

• Fractal Frogurt
• Posts: 434
##### Re: How many different ways are there to show such set ?
« Reply #10 on: November 23, 2020, 10:25:30 PM »
As field lines in exterior regions branch at each iteration boundary

Do you mean boundaries of binary decomposition ?

• Fractal Frogurt
• Posts: 434
##### Re: How many different ways are there to show such set ?
« Reply #11 on: November 23, 2020, 10:41:08 PM »
I have tried to recreate your cool image using
* boundaries of binary decomposition ( using Sobel filter) = bdb
* boundaries of level sets ( of escape time) = lc ( level curves)

so file name is lcbdb.png

Please note that I "normalize " parameters to have level curves cross at critical point ( z=0)

Code: [Select]
  ER_LSM = GiveER(20); // find such ER for LSM/J that level curves croses critical point and it's preimages
where
Code: [Select]
// find such ER for LSM/J that level curves croses critical point and it's preimagesdouble GiveER(int i_Max){ complex double z= 0.0; // criical point double ER; int i; ; // critical point escapes very fast here. Higher valus gives infinity for (i=0; i< i_Max; ++i ){ z=z*z +c; } ER = cabs(z); if (ER < 2 ) {printf("errror :  ER < 2.0 !!!!!\n");} return cabs(z); }
Of course your image is much better but the skeleton seem the same. Am I right ?

« Last Edit: November 25, 2020, 10:02:49 PM by Adam Majewski, Reason: https://github.com/adammaj1/Procedural-texture »

#### xenodreambuie

• Fractal Friar
• Posts: 105
##### Re: How many different ways are there to show such set ?
« Reply #12 on: November 24, 2020, 01:02:02 AM »
Thanks Adam; yes you are correct. The boundaries are the same as for binary decomposition. I was going to show an image with just the black lines, but decided to add a star to show how any pattern can be used.

The density and alignment of the grid depends on the high bailout, so I just adjust that visually for the desired result.

• Fractal Frogurt
• Posts: 434
##### Re: How many different ways are there to show such set ?
« Reply #13 on: November 24, 2020, 06:20:38 PM »

#### superheal

• Strange Attractor
• Posts: 94
##### Re: How many different ways are there to show such set ?
« Reply #14 on: November 25, 2020, 01:01:51 PM »
Xeno, can you elaborate a bit more on how you created this spiral like triangle inequality looking pattern? Do you mind providing some code?

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