Iterations III - Hopalong and Mira in variations

If you define x and y as screen coordinates and a, b and c as fixed values, then the iteration of Barry Martin's formula

xn = yn-1 - SQRT(ABS(b * xn-1 - c)) * SIGN(xn-1)
yn = a - xn-1

leads to the well known Hopalong patterns. This bahavior changes dramatically, if you put these formulas into the drawing algorithm of a Mandelbrot program.

The varied algorithm

If the parameters a and b are not seen as constants, but now - in a nested loop - used as screen coordinates, combined with the variable x as a color value depending on the number of iterations, you'll see new interesting patterns. These patterns seem to be a weird mixture of interfering ribbons and frost tracery. They don't look like typical fractal structures with its self-similarity. Every part of the drawing plane has its own pattern structures, which don't seem to be related with one another.

Detailed articles

Hopalong anders interpretiert

Hopalong varied

What is discribed above in a short manner, you can read much more detailed in a long article. But, sorry, you have to read it in German.

  1. Hopalong and the Mandelbrot set
    This article describes the origin of the images in the gallery and its imaginative interpretations.
  2. Program descriptions
    This article contains not only operating instructions, but also a comprehensive view of the algorithms.

Doenload programs

Three programs with the described variations are provided by Kurt Diedrich:

  1. Hopalong-Classic
  2. Hopalong-Special
  3. Mira-Special


The contents of this page were provided by Kurt Diedrich. If you have any questions you may ask the author (fraktalforschung <email symbol> tele2.de)


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