Poincare and the Mandelbrot Man

This is a Mandelbrot Set. Okay, actually it's a fractal produced using the Mandelbrot set of equations, but you know what I mean. I've heard it also called the Mandelbrot Man, (not to be confused with the Marlboro Man, who on occasion can be used to demonstrate chaos theory) so named because it looks a little like a roly-poly man. At any rate, this is created by a computer- a number is run through the equation and the result plotted on a graph, each point being assigned a different color due to its tendency within the equations. The Mandelbrot set is a non-linear equation- that is, it doesn't neatly solve itself into a line. Rather, we can zoom in on any point of the graph above (simply change the numbers we are using in the equation sets) and find a different pattern, yet this same figure repeats over and over. We can look at a miles-wide graph of the equation, or a single speck on the graph, and find first, that the same equation holds, and the pattern is the same across all scales.
So what? Well, the discovery of non-linear mathematics was a pretty important step, as it allowed for accurate models of real-world applications. Prior to this, mathematicians couldn't account for anomalies in a set of data. For any statistical or predictive purposes, they would use what was called a best-fit line- that is, they would come up with a line that best encompassed all the points on a graph, and call it good. It would, more or less, predict the action of any given point in time. However, exceptions and the occasional wild number would occur, and the best-fit line would not be able to predict it. This was at first termed chaos- the effects of chaos could not really be predicted, hence chaos.
However, as time went on, non-linear equations proved able to reveal a pattern within the seeming chaos. This was kind of noted in Poincare's Conjecture before the onset of non-linear mathematics. Poincare said that basically, the Earth could be flat, round or donut-shaped, and we would not know from walking on its surface. We would need to step back and view the surface from a higher level- in other words, in three dimensions rather than two, or four dimensions rather than three. The result would be that we are able to see the shape in its entirety, and better understand it. The same is true of mathematics- what may first appear as simple old chaos is due to a much more complex pattern, albeit one that is not readily available to our eyes. If we were to take five numbers and run them through the Mandelbrot set, we would see a bunch of dots spread out more or less at random. Take 300, and we would begin to see a pattern. Take one million, and then we can begin to see that the placement is not random at all- it fits perfectly into the pattern. Much like real life- we can't often see the pattern of the world, because of our point of view. Should we somehow be able to pull a Poincare, step back and examine the world carefully, we may well find that there is an underlying order in the chaos of modern life.