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Magnet Fractals
The "Magnet" fractals are based on a rather abstract model in theoretical physics that describes transitions between the magnetic and non-magnetic phases of certain types of magnetic materials. It turns out that an iterative process is necessary to work out what this boundary looks like, and that iterative process leads to these fractals. The shapes of these fractals correspond to the boundary between the magnetic and non-magnetic phases of the materials in this model.
The details of where the equation came from don't matter for the purpose of making fractal images. For more information, see The Beauty of Fractals by Peitgen and Richter, pp 129 et seq.
Part of the reason these fractals are not used more in either still art or video work is that they take much longer to calculate than the Mandelbrot set. The formulas for them are shown below.
| Magnet Type I | zn+1 = [(zn2 + c-1) / (2zn + c-2)]2 |
| Magnet Type II | zn+1 = [(zn3 + 3(c-1)zn + (c-1)(c-2)) / (3zn2 + 3(c-2)zn + (c-1)(c-2) + 1)]2 |
Compared to the Mandelbrot set formula, zn+1=zn2+c, these look like monsters! Especially type II.
Update January 2011: New deeper still images following in the wake of the erroneous magnet fractal deep zoom video published in November 2010 using the new Core i7 980X system.
Update April 2011: Montage1 video has a short non-deep zoom animation of the type I magnet fractal.
And this image here, which is, as far as HPDZ is aware, THE deepest image EVER of the type 1 Magnet fractal.
1000x800 1.4e-54 with 9X (3x3) stochastic supersampling. Render time: 6 hours 40 minutes.
   | In some areas, this fractal's formula, when iterated, behaves a lot like the Mandelbrot fractal formula z=z2+c, so we see tiny Mandelbrot sets scattered around within it. It is not strictly self-similar like the Julia set kind of fractals that come from Newton's method and its relatives, so the structure of the local neighborhood persists and gives rise to complex structures like the Mandelbrot set does. |
 1500x800 2.7e-26 |  800x800 9.6e-12 |  800x800 1.5e-26 |  800x800 2e-35 |
 500x500 3.5e-27 |  1200x800 1.5e-18 |  1200x800 6.2e-20 |  1200x800 3e-6 |
 1500x900 5.2e-12 |  1500x900 2.1e-13 |  1500x900 4.2e-14 |  1500x900 1.7e-15 |
 1500x900 1.1e-20 |  1500x900 4.5e-22 |  1500x900 1.4e-22 |  1500x900 1.7e-15 |
 1500x900 4.2e-16 |  1500x900 1.1e-16 |  15600x900 6.1e-18 |  1500x900 3.5e-19 |
 1500x900 1.9e-24 |  1000x800 3e-27 |  1000x800 1.5e-31 |  1000x800 4.9e-37 |
 1000x800 1.5e-29 |  1000x800 1.2e-30 |
  | The equation generating this fractal comes from a more complicated model of magnetic phase transitions than the type 1 fractal. Just like the type 1 Magnet fractal, this one has little Mandelbrot sets and a complex non-scale-invariant structure that is promising for zooming. |
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