Rated
Feb 21 2009
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1 review
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complex systems, fractals, mandelbrot set
• google.co.uk
Here is a thought experiment. It's tedious, but it leads somewhere truly wonderful. All you need (in your imagination) is an ordinary pencil, some coloured pencils, a pin, a very large piece of paper, something to draw a circle with, and something to measure with.
Draw a circle on the paper, large but leaving plenty of room outside it. Mark a light pencil dot anywhere inside the circle. Make a pin hole at the same place as the pencil dot.
Now make a second pin hole. You work out where to place it using a particular, very simple formula, based on where the pencil dot is and where the previous pin hole was (the same position, in this case). More about the actual formula later.
Keep doing this, placing more pin holes, whose positions are each calculated from where the pencil dot is, and where the previous pin hole was.
After a while, one of two things will happen. Either it will become obvious that all the pin holes are falling inside the circle, or one of the pin holes will fall outside the circle and thereafter all later pin holes will head right off the paper.
If the first thing happens, go back to the pencil dot and mark it in deep black. If the second thing happens, mark the pencil dot in some colour, the colour depending on how many pin holes it took to go off the paper.
Now start all over again with another pencil dot placed somewhere else, more pin holes, then colouring the pencil dot as before.
If you did all of this again and again until all your coloured pencil dots joined up, and your paper was big enough (of course your patience would run out and you would use a computer instead), you would see something like this:
This image of the complete Mandelbrot Set was created by Wolfgang Beyer with the program Ultra Fractal 3 - click the picture for higher resolution pictures and further details.
If you expand the view of the "valley" between the biggest blob and the next-biggest blob, you would see this:
By the time you have zoomed in another seven times you would see this in a tiny part of the original image:
And you can go on and on, deeper and deeper, uncovering still more wonders. I understand that some people, watching the computer program revealing the details, enter a kind of altered state of mind, and it isn't hard to see why.
When I struggled to find out what Mandelbrot's simple repeated formula actually meant and how it behaved, and where that wonderful complexity was coming from, there seemed to be two kinds of explanation. One kind was full of fearsome looking mathematical equations. The other kind avoided the maths but missed out on what was really happening.
So I tried to find a simple way of explaining it (for my own benefit) which didn't ignore the maths, but which didn't need someone to be a mathematician in order to understand it.
I don't know if I succeeded, but my best shot at it (should you be interested) is here.
You might have fun reading it - I had fun writing it, anyway!
