< Fractals < Iterations in the complex plane

Techniques:

"The golden rule behind most of my images is the following. Its aspects can be used to do pretty much everything that you find in my gallery: Given a location with a zoom level n, moving away from the center to a different center has the following effect: The shape at zoom level n is doubled at zoom level 1,5n in such a way that the rotational symmetry becomes 2-fold. At 1,75n the symmetry becomes 4-fold. At 1,875n the symmetry becomes 8-fold. ... In general: the zoom level increases in steps of 2^-1, 2^-2, 2^-3, ... and goes on forever. The symmetry increases by a factor 2 for every extra step. The limit of the sum of all of those steps 2^-1 trough 2^-n where n goes to infinity is 1, so after infinitely many steps we arrive at a finite zoom level. Indeed, at a depth of 2n, twice as deep as where we went off center, there is a small mandelbrot set, where the symmetry goes to infinity. The rule itself has not been proven as far as I know and there are endless exceptions where it is not exact. Sometimes shapes appear a little earlier than the rule would predict, but the small mandelbrot set will never occur FURTHER than 2n. (I think I know what the inaccuracy is, by the way.)" Dinkydau[8]

Video

References

  1. ↑ automated_julia_morphing by Claude Heiland-Allen
  2. ↑ Julia morphing symmetry by Claude Heiland-Allen
  3. ↑ fractalforums : towards-a-language-for-julia-morphing
  4. ↑ fractalforums : show inflection
  5. ↑ fractalforums inflection-mappings
  6. ↑ Causality in Fractals - shapestacking explained by Chillheimer Chillheimer
  7. ↑ Navigating to a Leavitt Embedded Julia Set by Robert Munafo'
  8. ↑ fractalforums : deep-zooming-to-interesting-areas/
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