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?

## Question on Mandelbrot Set

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### Re: Question on Mandelbrot Set

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:

x

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.

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:

x

_{n+1}= r x_{n}(1 - x_{n}), 0 <= x <= 1, 0 <= r <= 4Note 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.

### Re: Question on Mandelbrot Set

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 x

_{0}be a point on the boundary of the Mandelbrot set, and x be close to this point; that is, x = x

_{0}+ e

_{y}, 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.

### Re: Question on Mandelbrot Set

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.

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.

### Re: Question on Mandelbrot Set

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/

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

### Re: Question on Mandelbrot Set

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!

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### Re: Question on Mandelbrot Set

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

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

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