Sunday, December 27, 2009
As you most likely recognize, this is one of thousands of videos showing the amazing complexity of the Mandelbrot Set. This 'set', or collection of 2D numbers is created using one equation:
z --> z^2 + c
Where 'c' is the original number. If this series converges it is in the Mandelbrot set, if it diverges it is not. In other words, if you take a number, square it, and add that to the original number over and over and over will it go to infinity or not? 1^1+1 becomes 2, 2^2+1 goes to 5, 5^2+1 goes to 26 and this quickly runs up to infinity meaning that 1 is not in the Mandelbrot set. This set here is plotted in the complex plane, meaning real numbers are on the x axis (horizontal for those who haven't had math in a long time) and imaginary numbers on the y axis. Imaginary numbers being some multiple of 'i', or the square root of negative one. As a note this is a binary output, the different colors in the video (which based on this theory there should only be two) are based on how fast the number diverges.
This is not an explanation of mathematics, it's an example of chaos. One of the most famous examples of chaos from one of the best known names in the field. The larger point isn't the equation in and of itself but the simplicity of the equation. The above video shows us diving into the results of this equation and finding complexity at every level. Generalizations about behavior from a classical standpoint are rendered meaningless. The set never simplifies into generic groupings no matter how detailed your plot gets. This means that in a similar real world system predicting the outcome based on given input is rendered impossible. No matter how accurate your initial readings, this shows that even when hundreds of decimal places of accuracy is achieved, being off by that much can and will change how the system acts.
It's an interesting mix of determinism and unpredictability. Any 'chaos equation' is by definition determinate, it's an equation! Given a certain input there is a certain output. Now this does break down when non-linear differential equations are introduced as we don't really know how to solve those but that's not the point. The point is that laws don't have to be violated and results are repeatable numerically. The identical input can still give identical output so we become Calvinist in a sphere of predictable destiny. Yet identical input is a myth of binary technology.
In engineering achieving 5% error would be considered either luck or very good modeling (structural engineering anyways) in the hard sciences more precision is needed. Yet chaos proclaims that in systems under its control there is no level of precision sufficient to truly model outcomes. In fact that hard barrier of Schrodinger's uncertainty principle is hit and deterministic systems are made non-determinant by creating a fuzzy input even in reality. Bifurcations at miniature levels beneath the realm of even the theoretically measurable ensure that accurate predictions are rendered not merely impractical but truly impossible. The weather can only be predicted approximately three weeks in advance before quantum physics joins forces with chaos (sometimes referred to as 'highly non-linear systems') to render prediction unmanageable.
Yet we know climates and temperature variations over centuries. I may not know the high for three days from now but I can still predict the general temperature trend for July in 18 years with a rather surprising degree of accuracy. It gets warmed in the summer and colder in winter. This is the so called 'order in chaos' a prominent feature being 'strange attractors'. This shows that though any finite piece of data may be unpredictable the overall system has trends, has character. Patterns emerge not of true repetition but shapes begin to repeat. Lorenz's infamous figure eight, in the above video the two unequal circles that appear at the beginning make appearances through-out the infinite orders of magnitude. Phase space of so many equations show order emerging within the equations even when the data seems to have no discernible pattern. This is why chaos can be a science and not a permanent road block.
This also has no direct bearing on my life except that after reading Jame's Gleick's Chaos it i a subject that has occupied a very great deal of my time. What does it mean when it comes to plotting a course through life? What does it imply about our universe? Are there direct applications either to research I'm doing or to myself in general? I don't know but I've been thinking a lot about it. I may post more on the subject in which case I hope to use this as a springboard. But I may not, it's unpredictable. Chaos at work.