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        riddles >> hard >> Fractal Gemotry
        
(Message started by: simsimthedim on Aug 27^th, 2012, 9:22am)

Title: Fractal Gemotry
Post by simsimthedim on Aug 27^th, 2012, 9:22am

I saw a documentary on fractal geometry. The person who uncovered this math is Mandelbrot. I have some questions I hope can be answered:

- This documentary was PRO Mandelbrot and fractals and I never heard opposing views. Are there any within the math community?

- If Mandelbrot wrote the hard code for these fractals - how can they be tested proven or disproven?

- The only benefits mentioned were possible gaming theory applications. Are there others?

- Why didn't Mandelbrot receive some sort of Nobel prize?

- Is fractal geometry required math in secondary education?

Title: Re: Fractal Gemotry
Post by towr on Aug 27^th, 2012, 9:55am

on 08/27/12 at 09:22:58, simsimthedim wrote:

- This documentary was PRO Mandelbrot and fractals and I never heard opposing views. Are there any within the math community?

What sort of opposing view could there be? Fractals are a mathematical reality. It's not an opinion.
I would note, however, that fractals had been discovered before Mandelbrot started working on them, for example the Koch snowflake was published 71 years prior to Mandelbrot naming them fractal. But he did a lot of work on examining their properties.

Quote:

- The only benefits mentioned were possible gaming theory applications. Are there others?

You can use them for image compression (though that's related to gaming applications). You can also use them for password systems. There's also art.
There's probably more applications than you imagine, certainly more than I can come up with on the spot. [edit]Indeed, http://en.wikipedia.org/wiki/Fractal#Applications_in_technology [/edit]

Quote:

- Why didn't Mandelbrot receive some sort of Nobel prize?

There is no Nobel prize for mathematics.
He has won numerous prices and honors, but I don't know if any of them are for his work on fractals.

Quote:

- Is fractal geometry required math in secondary education?

No; probably not even if you do a maths degree, unless you specialize in a related area (like perhaps topology).

Title: Re: Fractal Gemotry
Post by Noke Lieu on Aug 27^th, 2012, 7:27pm

on 08/27/12 at 09:22:58, simsimthedim wrote:

- If Mandelbrot wrote the hard code for these fractals - how can they be tested proven or disproven?

I'm not sure I understand what you're asking here.
The Mandelbrot set, which is the factal that most people recognise, is essentially a way of graphing an unusal equation.
1. You need to choose a number, and keep a running total.
2. Starting your running total at 0.
3. Square your total, and add the number you chose.
4. That gives you a new total.
5. Goto step 3.

If the total grows and grows, then the number you chose isn't included in the graph. If it doesn't grow and grow (it might get closer and closer to a particular number, it might jump between two numbers) then it does get included.

People came up with this idea before Mandelbrot, but he was (AFAIK) the first guy to think of making a graph out of it. (Which isn't as easy as it might sound...)

Quote:

- The only benefits mentioned were possible gaming theory applications. Are there others?

LIke Towr mentions, yep there are many. It doesn't really help you check you've got the right change at the shop, but it's not just esoteric picture making...

Quote:

- Is fractal geometry required math in secondary education?

Depends what you mean by 'required'.
As an example of beauty that's inherent in maths, then it's a pretty good idea. As a bit of mind bending, counterpoint fun, (I'm thinking of examining Koch Snowflake area versus perimeter) to allow discussion of the approaches to finding areas... certainly.
As an assessable element? Probably not.

Title: Re: Fractal Gemotry
Post by rmsgrey on Aug 28^th, 2012, 5:48am

My preferred way of understanding the Mandelbrot set is one I found in Arthur C Clarke's The Ghost from the Grand Banks (a novel about an attempt to raise the Titanic, but that's incidental)

If you take a number and square it, then square the result, and square that, and so on - something you can do with any decent pocket calculator (or the Calculator program that comes free with Windows) then, for almost all numbers, one of two things happens: either the result grows until the calculator gives up; or it shrinks until the calculator gives up. If you shade the number line red where the numbers explode, and blue where the numbers collapse, then you end up with two points left (green, say) at -1 and 1 - the boundary between exploding and collapsing.

If you do the same thing with complex numbers, you end up with a fairly boring green circle with red outside and blue inside.

What Mandelbrot did was, instead of just squaring every time, square, then add the original number, so if you start with -2, you get -2 squared is 4, add -2 gives you 2, and the next step, 2 squared is 4, add the original -2 gives you 2 again. So -2 is one of the boundary points between collapsing to 0 and exploding to infinity. For any positive real number, or any negative number below -2, you always explode to infinity, and for numbers between -2 and 0, you collapse to 0. Doing it with complex numbers gives you a fantastically complicated boundary shape - and that boundary is the Mandelbrot Set.

The colours you often see in images of it come from counting how many times you need to square-and-add to decide that that point escapes and will explode to infinity - a simple test being if any iteration falls outside the circle of radius 2 around 0. If you need to square-and-add fifteen times to get outside that circle, then colour the original point with the fifteenth colour...