WTF by chaos5

From a post FractalMonster left in my deviation Ad Nauseum, it occurred to me that I could expand the polynomial form of a cubic Mandelbrot.

This is a rendering of Z->Z^3-2Z^2+Z
Now looking at it, I could swear it just morphed into a Julia set! I can not explain this, and was hoping for some feedback about this phenomenon.

Thank you!

$:iconfractalmonster:$

FractalMonster Featured By Owner Aug 14, 2007

Just arrived from job

Because I recognized your deviation I starded Ultrtra Fractal, formula CubicParameterspace3 which draw the standard form z -> z^3 - 3a^2 z + b. I put a = 0.66 + 0i, b = #pixel. Here comes the illustrations:

[link] z = +a, forming the set M+,
[link] z = -a forming the set M-.
[link] CCL (Cubic Connectedness Locus) are the set which his common to M+ and M-, and
[link] CCL + SetBorders making the whole counturs of both sets visible

The four illustrations have exactly the same coordinates. They talk for themselves

chaos5 Featured By Owner Aug 14, 2007

Awesome illustrations!!
I had noticed that it fliped with the +/- but it had not occured to me how they could line up when overlayed.

Thanks for psoting

$:iconfractalmonster:$

FractalMonster Featured By Owner Aug 15, 2007

Thank you Alister

Exactly

In fact there is only one layer, and "the layering" works in the following way (now I make a citation from article 19b, page 4, in the chaotic series [link] :

"9) Under ”M”, select CCL and behold ”SetBorders” (figure 9). If only one of the two critical points has a bounded orbit, UF will color the pixel corresponding to that coordinate after the number of iterations requiring for the other critical point to escape. If, on the other hand, both critical points escape, the pixel representing that coordinate is colored after the number of iteration of the critical point that escapes last (has the highest iteration-number)."

But at most I draw these type of images in two layers, one for M+ and the other for M-. This makes it possible to distinguish the two sets from each other by using different gradients. Moreover one get faster calculations. Bur all of this can be read of in especially article 19b

Regarding Cubics you have a quick introduction in [link]

Regarding Pentics, which has 4 sub sets check an compare [link] , [link] , and [link] , all having the same coordinates

$:iconfractalmonster:$

FractalMonster Featured By Owner Aug 14, 2007

Looks rather autentic

Note that there are 2 critical points for cubics. The very right dark part is situated inside the other subset. It may be interesting to perform some zoom in that part