Fractals/Iterations in the complex plane/sculpt

   Julia morphing, an artistic Mandelbrot set zooming technique, gives angled internal addresses that end with a regular structure like:
   
 $...p\xrightarrow {1/3} 2p+k\xrightarrow {1/3} 2(2p+k)+k...$
 
   (...) Embedded Julia sets are (relatively) surface features with simple angled internal addresses, while Julia morphs are much deeper and the addresses are more complicated.

Techniques:

• Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) = "using the property that zooming in towards baby Mandelbrot set islands doubles-up (and then quadruples, octuples, ...) the features you pass. This allows you to sculpt patterns, here the pattern has a tree structure."
• Inflection
• shape stacking
• Navigating to a Leavitt Embedded Julia Set
• minibrots
• symmetry
• repetition
• Letteres in the Mandelbrot set

"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

inflection

inflection mapping = translation and squaring of complex coordinates before regular Mandelbrot or Julia set iterations

Patterns

Patterns are sculpted in the intricate shape of the boundary the Mandelbrot set when zooming ( especially deep zooming). Zooming are reflected in the complex dynamics, in particular in the binary expansions of the pairs of external rays landing on the cusp of each baby Mandelbrot set copy at the centre of each phase.

Links

• Patterns in deep Mandelbrot zooms A simple formula leads to emergent complexity. Paper by Claude Heiland-Allen and www page

Video

• FractalNet HD - Morphing Julia set 1 Michael Hogg
• Natural shapes discovered in the Mandelbrot set fractal by logicedges