Some general questions about the Mandelbrot Set (Fractals)

[Kyriakos]

Some very general questions:

1) Why was this particular polynomial set as the one iterated?


z_{n+1}=z_n^2+c

(http://en.wikipedia.org/wiki/Mandelbrot_set)

2) I read that the Set, with the characteristic shape ( http://en.wikipedia.org/wiki/File:Mandel_zoom_00_mandelbrot_set.jpg ) is the collection of all the points for which it still holds true that the series created by the iteration of X (ie x, F(x), F(F(x), F(F(F(x) and so on) will remain bounded, which means that it will not escape to infinity. I can see why the series (which involves complex numbers) remaining bounded is significant, but what is the special math significance of an iterated series regardless of complex numbers or not?

*

Thanks in advance for any help with the two questions. I suppose i will read more of this, but the first question is by far the one i mostly would like a specific answer to

 

[nc-1701]

Some very general questions:

1) Why was this particular polynomial set as the one iterated?


z_{n+1}=z_n^2+c

To some degree this is arbitrary... Very many sequences get studied in mathematics, when one with a cool property or other significance is studied it gets remembered, obviously generating the Mandlebrot set counts as having a cool property.
Now obviously that isn't a great answer, because it doesn't illuminate very much, but I'm not sure I have a better one. In general squares tend to behave "well" in the Complex numbers, since the complex numbers are the splitting field of the quadratic polynomial x^2+1. I'm probably missing the point of your question somehow, perhaps you could rephrase it in the context of my answer to let me understand it better.

<snip>...but what is the special math significance of an iterated series regardless of complex numbers or not?

Sequences and series are one of the major objects studied in math particularly in the field of analysis. Historically you could say this was motivated by the creation of calculus which requires us to have a working (for arithmetic purposes) concept of infinity (or rather division by zero, but meh), which to some degree is achieved by using sequences/series.
In general they are a complicated and interesting mathematical object, that is very accessible, after all it's just a bunch of numbers. Recursive sequences, like this one, are interesting because they are easy to define, and tend to have interesting properties.
In general sequences being bounded (and where they limit too) is the standard question to ask about a sequence, but that part isn't surprising.

 

[Kyriakos]

Thanks

I had asked the same in some other forum, and got the 'let-down' answer as well (that the specific polynomial seems to have been chosen as the one to be iterated largely out of practical reasons- older computers).

I had hoped that the polynomial played a more crucial role in the whole generated property of a bounded iterated series including complex numbers. Maybe it does (?), since you mentioned the link between squared parts in polynomials and complex numbers (which i might read more about).

Thanks again

 

[nc-1701]

Thanks

I had asked the same in some other forum, and got the 'let-down' answer as well (that the specific polynomial seems to have been chosen as the one to be iterated largely out of practical reasons- older computers).

I had hoped that the polynomial played a more crucial role in the whole generated property of a bounded iterated series including complex numbers. Maybe it does (?), since you mentioned the link between squared parts in polynomials and complex numbers (which i might read more about).

Thanks again

There are absolutely lots of polynomials you could do this with.
z_n+1 = z_n^2 +z_n + c
Comes to mind as an obvious example, the question is what does the corresponding set look like. In the case of the Mandelbrot set we get a set that is super cool, and closely resembles a fractal. However with some other polynomial we may get a set that looks "boring" i.e. a circle or something.

If you scroll most of the way down your link and read "generalizations" it has a little bit about other mappings and a gif that shows some of them. The gif shows how it changes as you raise the power from 2 to 5, and demonstrates how exponentiation acts as a rotation in the complex plane.

 

[Kyriakos]

Thanks again...

After a little more reading, i found this:

The progression of which just looks amazing, particularly the rapid turn in the scaling before the nest-like shape is formed Fractal comes from Fractus (i read), so i suppose it would be equivelant to the Greek term &#929;&#942;&#947;&#956;&#945; (Regma), which signifies a natural break in a surface.

 

[Gigaz]

The progression of which just looks amazing, particularly the rapid turn in the scaling before the nest-like shape is formed Fractal comes from Fractus (i read), so i suppose it would be equivelant to the Greek term &#929;&#942;&#947;&#956;&#945; (Regma), which signifies a natural break in a surface.

I think the name refers to the dimension of the edge of the fractal.

A common object has n dimensions and an edge with n-1 dimensions. A sphere has 3 dimensions, but the surface is two-dimensional.

The fractal in dimension n, however, has an edge with a Haussdorf dimension. It is usually n-1 + (a fraction<1). The fractal is called a fractal because of it's fractional dimensionality.

The edge of the Mandelbrot set is an exception, by the way. It's edge is two-dimensional and has an area of approximately 1.5 (length units)².

 

[Kyriakos]

^I will read more on that too, i did pick up the dimensional theme in 'fractals' and that they do not have an integer as a number of dimensions, but currently i have not read any elaboration on what surely is a very interesting quality to learn about in the math model