6/28/2009

TRIVIA QUESTION: MANDELBROT SET

MANDELBROT SET
Question: How is evolution by natural selection like a fractal?


No, not Fraggles!

Fractals. To understand my trivia question, you will need to know a little something about fractals, such as the mathematical object known as the Mandelbrot Set.

The technical definition (somewhat elaborated in the video below) goes something like this: the Mandelbrot set, M, consists of the points produced by a set of complex quadratic polynomials that do not 'escape' to infinity, where z always remains equal to or less to some value set by some complex parameter, c, as follows:


Gobbledygook to most of us, but it may help if you realize that the absolute value of any particular working out of the set of points (what we might call an iteration) would look like this:


This expression (which I'll write P* for convenience) has two dimensions, and so it can be worked out that the above expression can be worked out in a disk (the flat, two-dimensional surface bounded by a circle). This closed disc by definition has a radius of 2 arbitrary units. Thus, if the value of the expression ever exceeds 2, then the set of points diverges to infinity and is no longer part of the set. This leads to drawings like this (courtesy of this Wikipedia article), where all the black parts represent the set of points where P* is less than 2, and the white parts the set of points where P* is greater than two (and thus diverges to infinity). See?

Objects like this have a remarkable property of self-similarity at different levels of scale. Kind of like much of the natural world, which is a key to my little trivia question.




But why get hung up on mere symbols, when visual representations accompanied by a (highly) original song can supplement our understanding?



If you like the tune by Jonathan Coulton, then visit his web site and buy his music!

1 comment:

CarlaCarlaCarlaCarla said...

It's Red! I love Red!

I think you'll understand that I didn't really read this blog entry, but I certainly noticed Red's charming photo. Thanks for including it.