## will every mandelbrot zoom end in a black point?

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

zukenft
Posts: 44
Joined: Mon Dec 20, 2010 11:34 am UTC

### will every mandelbrot zoom end in a black point?

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?

Xenomortis
Not actually a special flower.
Posts: 1409
Joined: Thu Oct 11, 2012 8:47 am UTC

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

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

gmalivuk
GNU Terry Pratchett
Posts: 26023
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

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

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.)
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

zukenft
Posts: 44
Joined: Mon Dec 20, 2010 11:34 am UTC

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

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

### Who is online

Users browsing this forum: No registered users and 7 guests