Sequence Fractals Part II #13


si = a,-a,…; a = -0.718
Center: M1,1.00; Zoom = 5
Start: 0
Julia / dynamic / variable view.

Going back to critical point 0 as the start point, and switching to Julia view. Compare to Sequence Fractals Part II #9. This one appears connected, but it is not. Under deep zoom there is a small gap on the center where the two shapes come together. This is expected as M1,1.00 is a repelling fixed point after one iteration. Note however there are large connected components.

Compare the Julia pictures and also compare the parameter space pictures M1,1.00 in Sequence Fractals Part II #8 (a=-0.179) and Sequence Fractals Part II #11 (a=-0.718). In the first picture of each set (a=-0.719) imagine M1,1.00 moving through a field of settled dust, in its wake it kicks up the dust produces the colorful swirlies. Yes, water would be a better analogy, but I need dust for the next part. Now imagine in the second picture (a=-0.718) there is a vacuum cleaner nearby. It sucks up most of the dust that gets disturbed, exposing the clean white table underneath.

I think the analogy comes pretty close to the actual dynamics. Points near a repelling cycle or pre-cycle will tend to dance around a lot before heading off to infinity. This is exactly the colorful spirals that make fractal pictures so interesting. If there is an attracting basin nearby, such as the attracting fixed point associated with some other critical point, some will get sucked in during the dance.

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