Mandelbrot

[Thomas(believer)]

I suppose that most of the readers here have heard of the Mandelbrot set
and have seen the beautiful pictures of it.
For those who have not, here is a link: http://en.wikipedia.org/wiki/Mandelbrot_set

The basis for these beautiful and complex pictures is a very simple mathematical equation: z_{n+1} = {z_n}^2 + c. where z and c are complex numbers.

I have a question about this and would like to hear your opinions.

Suppose we will ask the most brilliant mathematicians on earth to solve a puzzle.
We show them a picture of a set like the Mandelbrot set and their task will be to discover the mathematics behind this picture.
Can they solve this puzzle or would that be too difficult?

[Vaelroth]

Assuming that every mathematician worth his salt has at least read about the Mandelbrot set, I would assume that they would either come up with the equation on their own and work out the variables when presented with a graph of the Mandelbrot set.

[Disinfo Agent]

I have a question about this and would like to hear your opinions.

Suppose we will ask the most brilliant mathematicians on earth to solve a puzzle.
We show them a picture of a set like the Mandelbrot set and their task will be to discover the mathematics behind this picture.
Can they solve this puzzle or would that be too difficult?I think you need to explain well what you mean by "show them a picture of a set like the Mandelbrot set", and "discover the mathematics behind this picture".

P.S. Thought a trivial reply is, of course, that most mathematicians are well acquainted with Mandelbrot sets, and their graphs. :D
They may even know about Julia sets (http://en.wikipedia.org/wiki/Julia_set).

[Thomas(believer)]

I mean not any of the known sets. So no Manelbrot set, no Julia set etc.

[01101001]

I mean not any of the known sets. So no Manelbrot set, no Julia set etc.

The question needs serious firming up.

What are you revealing, a subset of the infinite set of function values? And, then you want some most-simple-form function that will yield those known values and all other hidden values over an infinite domain?

If so, I'd say impossible, because the unknown values could be anything.

For instance, if I showed you all the values, for the domain -1000 to 1000 for:

f(x) = if x is 1000000, 1; otherwise 0

How could the contestant possibly know the function isn't f(x) = 0 or something equally simple but wrong? Not interesting.

On the other hand, if you were to specify all the possible forms of the function, and a maximum length of the function expression, one could in theory (but perhaps not practically) simply enumerate all possible functions up to the maximum length, evaluate them over the revealed domain and list exactly the ones that fit the revealed values. That's boring.

Your question is too mushy.

[Disinfo Agent]

There are statistical methods for fitting a function to a set of data. Usually, though, they don't look for the 'right' function, but for one that is 'close enough', and not too difficult to handle. Also, you have to make some initial restrictions on the shape of the curve.

Curve fitting. (http://mathworld.wolfram.com/CurveFitting.html)

[hhEb09'1]

The basis for these beautiful and complex pictures is a very simple mathematical equation: z_{n+1} = {z_n}^2 + c. where z and c are complex numbers.I want to stress that that is a basis, not the generating equation, as we normally think of a graph of an equation. When that equation is applied to a point in the plane, and then again to that answer and repeated over and over forever, if it does not grow to infinity, then the point is said to be a member of the Mandelbrot set. It is totally impossible to show anything but an approximation to a "picture" of the Mandelbrot set. This is not the same thing as being constrained by the "width" of lines, where convention imagines it to have no width. There are just an infinite number of branches to the set that cannot be represented graphically.

[gwiz]

Or is this thread a subtle way of undermining the complexity probability arguments of Intelligent Design?

[Thomas(believer)]

You could say the question is: Is it possible to find the "simple" idea behind a complex observation. The Mandelbrot-set seemed to be a good and illustrative example.
I know this board is about astronomy, but I don't want to start with the entire universe right away.:p

[publiusr]

Some of those fractal artworks make my head hurt.