highlighting parts of the Mandelbrot set in renders

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Online claude

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« on: November 03, 2018, 06:54:35 PM »
In this thread I will show various ways to highlight different parts of the Mandelbrot set in renders.

First up, minibrots (and other mu-units) via renormalization.

The idea is to see when f_c^p(0) escapes a small region around 0 (for other formulas, replace 0 with a critical point).  This is explained more thoroughly in Demo 5 of Mandel 5.15 by Wolf Jung. The software is GPL but I couldn't decipher it (too much mutable state and magic values), consider the code on the linked web page a public domain implementation.

0. set ER2 = 4
1. find the nucleus C and period P of the owning atom, for example by Newton's method
2. find the d/dz of the nucleus' periodic cycle starting from Z=C (the product of 2Z excluding the Z=0)
4. set er2 = 4 / |d/dz|^2
5. iterate
Code: [Select]
    double _Complex z = 0;
    double _Complex dc = 0;
    double de = -1;
    double DE = -1;
    for (int k = 0; k < N; ++k)
    {
      if (cnorm(z) > er2 && de < 0)
        de = sqrt(cnorm(z)/er2) * log2(2*cnorm(z)/er2) / (cabs(dc) * pixel_spacing);
      for (int l = 0; l < period; ++l)
      {
        if (cnorm(z) > ER2 && DE < 0)
          DE = sqrt(cnorm(z)/ER2) * log2(2*cnorm(z)/ER2) / (cabs(dc) * pixel_spacing);
        dc = 2 * z * dc + 1;
        z = z * z + c;
      }
      if (de >= 0 && DE >= 0) break;
    }
    double MU = tanh(fmax(DE, 0));
    double mu = tanh(fmax(fmax(de, DE) * 4 / (period * period), 0)); // this 4/p^2 scale factor is preliminary, needs more research!
    colour = mix(red, mix(black, white, MU), mu);

full code and more example images at: https://mathr.co.uk/mandelbrot/mu-unit/
the attached shows a period 9 island in the antenna of the 1/3 mu-unit

I modified the distance estimation formula to avoid glitches near iteration band boundaries in the far exterior.  Seems to work, but I'm not sure if it will break in other views.

I suspect the 4/period^2 factor should really be "1 / feigenbaum constant ^ log2(period)" for period double bifurcations?  and other sequences of multiply islands etc might benefit from different factors?  To be researched!

References:
http://mndynamics.com/indexp.html
http://www.mrob.com/pub/muency/muunit.html

Online claude

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« Reply #1 on: November 04, 2018, 03:19:43 PM »
Next, mu-atoms aka hyperbolic components via multiplier map:

If |z| reaches a new minimum at iteration p, find the limit cycle z_0 = f_c^p(z_0) by Newton's method.  c is in a hyperbolic component of period p when |d/dz f_c^p(z_0)| <= 1.

Code: [Select]
    double _Complex z = 0;
    double _Complex dc = 0;
    double M = 1.0 / 0.0;
    double DE = -1;
    int highlight = 0;
    for (int k = 1; k < N; ++k)
    {
        dc = 2 * z * dc + 1;
z = z * z + c;
        if (cnorm(z) < M)
        {
          M = cnorm(z);
          double _Complex w = z;
          double _Complex du;
          for (int l = 0; l < 30; ++l)
          {
            double _Complex u = w;
            du = 1;
            for (int m = 0; m < k; ++m)
            {
              du = 2 * u * du;
              u = u * u + c;
            }
            w -= (u - w) / (du - 1);
          }
          if (cnorm(du) < 1)
          {
            DE = 0;
            highlight = k == period;
            break;
          }
        }
        if (cnorm(z) > ER2)
        {
          DE = sqrt(cnorm(z)/ER2) * log2(2*cnorm(z)/ER2) / (cabs(dc) * r0 / H);
          break;
        }
    }

More images and full code at https://mathr.co.uk/mandelbrot/mu-atom/

(In fact the d/dz is a conformal map from the mu-atom to the unit disc, this can be used for other colouring algorithms.)

Offline Adam Majewski

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« Reply #2 on: November 04, 2018, 05:25:52 PM »


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