Question on Mandelbrot Set

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

Lawsome
Posts: 86
Joined: Mon May 07, 2012 7:35 pm UTC

Question on Mandelbrot Set

Postby Lawsome » Sun Sep 28, 2014 8:09 pm UTC

This is a ridiculously specific (and yet somehow very vague) question to which there may not be an answer, but it's been bugging me for ages.

We all appreciate the boundary to the Mandelbrot set is a fractal, but WHY is it a fractal? Why should it be that the iterative process used to create it leads to all of these nested Julia sets, and self-similarity, etc that makes it so awesome. Do we really know?
Spoiler:
Image
Quizatzhaderac wrote:
Rosewinsall wrote:DOWN WITH CERTAINTY!

Are you certain of that?

Moose Anus wrote:I let my wife think I'm watching porn in the bathroom late at night, but I'm really playing Dwarf Fortress instead.

User avatar
PM 2Ring
Posts: 3713
Joined: Mon Jan 26, 2009 3:19 pm UTC
Location: Sydney, Australia

Re: Question on Mandelbrot Set

Postby PM 2Ring » Mon Sep 29, 2014 6:53 am UTC

Why not? :)

On a more serious (but still very informal) note, it seems reasonable to me that iterative processes / recurrence relations lead to self-similar structures. The iteration acts like a kind of filter that only a self-similar structure can pass through.

When the recurrence relation is linear, the resulting structure is fairly tame: a straight line is a linear self-similar structure. But if we make the recurrence relation non-linear, then things can start to get interesting. One of the simplest non-linear recurrence relation is the logistic map:

xn+1 = r xn(1 - xn), 0 <= x <= 1, 0 <= r <= 4

Note that for some values of the parameter r the behaviour is relatively boring, but for others things get gnarly. :)

I know that doesn't really answer your question, but I hope that it's a start. :)

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

Re: Question on Mandelbrot Set

Postby Elmach » Wed Oct 01, 2014 5:50 am UTC

Lawsome wrote:Why should it be that the iterative process used to create it leads to all of these nested Julia sets[...]?

I don't (think I) know the answers to the other questions, but I do know the answer to this one. (I don't know where I found this from, though.)

Let x0 be a point on the boundary of the Mandelbrot set, and x be close to this point; that is, x = x0 + ey, where e is really small and Greek.

It is trivial to show (read: I forgot how this is done, but it should be easy to someone who is not half-asleep) that for sufficiently small e, x is in the Mandelbrot set iff y is in the Julia set of x_0, in some vaguely informal way. (EDIT: See the link to baez's page two posts down, scroll to the part about arrows -- roughly the same concept)

Thus, the Mandelbrot set is a collage of tiny copies of the Julia sets.

Since the Julia set is fractal, the Mandelbrot set is fractal. (citation needed)
Last edited by Elmach on Wed Oct 01, 2014 6:21 pm UTC, edited 1 time in total.

User avatar
PM 2Ring
Posts: 3713
Joined: Mon Jan 26, 2009 3:19 pm UTC
Location: Sydney, Australia

Re: Question on Mandelbrot Set

Postby PM 2Ring » Wed Oct 01, 2014 10:37 am UTC

Here's a visual demonstration of what Elmach's talking about; a Mandelbrot collage consisting of a bunch of Julia set images.

http://math.ucr.edu/home/baez/696px-725_Julia_sets.png

Thanks, skullturf . I couldn't remember where I'd first seen this image.
Last edited by PM 2Ring on Sat Oct 04, 2014 4:56 am UTC, edited 2 times in total.

skullturf
Posts: 556
Joined: Thu Dec 07, 2006 8:37 pm UTC
Location: Chicago
Contact:

Re: Question on Mandelbrot Set

Postby skullturf » Wed Oct 01, 2014 5:18 pm UTC

This might not answer the question exactly, or succinctly, but it's an exploration by John Baez of "why" certain processes lead to fractal sets.

http://www.math.ucr.edu/home/baez/roots/

Lawsome
Posts: 86
Joined: Mon May 07, 2012 7:35 pm UTC

Re: Question on Mandelbrot Set

Postby Lawsome » Fri Oct 03, 2014 10:09 pm UTC

Thank you guys this information has been incredibly useful! Elmach I absolutely love that proof, and skillturf that's an awesome article. Thank you guys a ton! :D
Spoiler:
Image
Quizatzhaderac wrote:
Rosewinsall wrote:DOWN WITH CERTAINTY!

Are you certain of that?

Moose Anus wrote:I let my wife think I'm watching porn in the bathroom late at night, but I'm really playing Dwarf Fortress instead.

User avatar
Girl-With-A-Math-Fetish
Posts: 205
Joined: Mon Jun 30, 2014 9:02 am UTC
Location: The Complex Plane in the Mandelbrot Set
Contact:

Re: Question on Mandelbrot Set

Postby Girl-With-A-Math-Fetish » Sun Oct 12, 2014 1:09 am UTC

It's simple!
It's defined by an iterative process, as you said, and it exhibits principles of self-affinity, and it is not differentiable at any point, and such :V
I'm sexually attracted to the International Space Station. You have now had your daily dose of Internet.


Return to “Mathematics”

Who is online

Users browsing this forum: No registered users and 15 guests