will every mandelbrot zoom end in a black point?

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zukenft
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will every mandelbrot zoom end in a black point?

Postby zukenft » Mon Apr 17, 2017 10:34 am UTC

was into mandelbrot zoom videos recently. the good ones always end with a 'mini-brot' after a huge amount of zoom, so it got me thinking, will this always be the case? or is it just an already known point inside the set that yield good views during the zoom in?

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Xenomortis
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Re: will every mandelbrot zoom end in a black point?

Postby Xenomortis » Mon Apr 17, 2017 11:00 am UTC

I don't think zooming into the point (0+0i) will ever yield an "interesting" zoom.
I suspect the points chosen for fancy zooms are "special".
https://en.wikipedia.org/wiki/Mandelbro ... similarity
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Re: will every mandelbrot zoom end in a black point?

Postby gmalivuk » Mon Apr 17, 2017 12:50 pm UTC

Every point on the boundary of the Mandelbrot set is near something interesting, so you don't need to pick "special" points except to the extent that you need to pick boundary points (because obviously zooming into the center or someplace well outside the set isn't going to be interesting).

Xenomortis wrote:I don't think zooming into the point (0+0i) will ever yield an "interesting" zoom.
I suspect the points chosen for fancy zooms are "special".
https://en.wikipedia.org/wiki/Mandelbro ... similarity
That says it's self-similar in neighborhoods of the Misiurewicz points, which are dense on the boundary of the Mandelbrot set.

If we define "interesting" as meaning (or at least including) "eventually gets to a mini-brot", then every point on the boundary has interesting points arbitrarily close to it.

(That said, it may make it subjectively more interesting if the mini-brot at the end is sort of a surprise, which would require zooming on a region that doesn't have obviously self-similar features at larger scales.)
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zukenft
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Re: will every mandelbrot zoom end in a black point?

Postby zukenft » Mon Apr 17, 2017 7:19 pm UTC

ah, so my suspicion is half-correct then? you just have to move the center of zoom a little bit after you see one minibrot near?
if so, i guess making one of those zoom videos could be automated with twice the computing time. choose a random point and magnification, move the center coordinate when there's a black point near after the desired zoom level, then make a new time lapse with the new center, now on a minibrot :D


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