### 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 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.

- Hopalong and the Mandelbrot set

This article describes the origin of the images in the gallery and its imaginative interpretations.
- 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:

- Hopalong-Classic
- Hopalong-Special
- Mira-Special

Credits

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)