Monday, June 15, 2015

Our Amazing Universe: The Mandelbrot Set

While visiting with Dan Hansen a few months back I was introduced to the incredible concept of the Mandelbrot Set. It was a concept I was unfamiliar with, and in the event that you are as well here's how Wikipedia begins its explanation:

The Mandelbrot set is the set of complex numbers 'c' for which the sequence ( c, c² + c, (c²+c)² + c, ((c²+c)²+c)² + c, (((c²+c)²+c)²+c)² + c, ...) does not approach infinity. The set is closely related to Julia sets (which include similarly complex shapes) and is named after the mathematician Benoit Mandelbrot, who studied and popularized it. Mandelbrot set images are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.

After briefly detailing the rules regarding Mandelbrot sets, Wikipedia closes its intro with these statements.

Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization.

Who says mathematics can't be fun? You can read the full explanation and history of this fractal at this page on Wikipedia.

It was the YouTube video that brought home how amazing the Mandelbrot set phenomenon really is. As you zoom in on one arm of the set you discover that the set is replicated within the set, ad infinitum.

While watching the above video (you can turn the sound off if you think the music is cheesy) I began to wonder again what one could discover by zooming in on atoms, the building blocks of our material world. Since my high school chemistry days we've made incredible advances in the exploration of smaller and smaller particles. How small does it go?

Of course on the other end of things, how big does it go? The universe, that is. How did this happen? And what does it all mean?

That's enough for a Monday morning, I think. It's awesome!

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