Original Post Date: Jun 20, 2014

Original Post: http://bit.ly/1DjhRCZ

I don’t know what else to add. This is one of my posts where I pretty much have my own ideas. Well, I guess I should have mentioned the Three-Body-Problem, and more about Godel’s Incompleteness Theorem…

# The Beauty of Chaos and Mathematics

Mathematics is not itself about numbers and symbols. Those numbers and symbols are a part of the grand metaphor for the absolute truth of what is mathematics.

Given two particles of identical mass that only but gravitationally interact with each other, it seems obvious how they’d mathematically interact. The force that draws them together is simply Newton’s gravitational constant times both of their masses divided by the square of the distance between them. It’s nothing more than hardly high-school physics. An infinitesimal of time ticks by and the particles move accordingly. They have a new position and a new trajectory. The function is applied again, and the particles move accordingly, again, with a new trajectory. This process goes on and on.

It still appears to be a simple system. After all, with two particles, it seems that complexity is completely nonexistent in the system. It might seem complicated to a layman, but it is not complex, by definition—nor is it complicated to any high school student with a basic understanding of physics and math. In fact, this system is a far less complex system than reality; reality would include far more forces than just the gravitational force, and far, FAR more particles. Either way, this particle system seems rather simple.

Will it follow a predictable pattern? It might seem like it does at first glance. That is the beauty of mathematical chaos. This one simple pattern follows a seemingly unpredictable nature, even though it is defined by one simple function. This Lorenz Attractor system is a good example of mathematical chaos. If we change any of the parameters by just a tiny bit, no matter how small, given an enough amount of time, the systems appear to have no similarity in their trajectories. It’s simply beautiful.

Throw in a third particle. How about a hundred particles? Maybe we’ll throw in the few googols or so of subatomic particles that we know of into the mix, and all of the strange ways they behave, just to add to the complexity. The beauty is that these systems are theoretically governed by a set of a few mathematical rules. Whether we have discovered these patterns or not yet, these patterns do exist.

Gödel’s Incompleteness Theorem states that there might be problems that we will never be able to solve. From a philosophical sense, this is true. For all we know, solipsism could be the truth, and everything is but a mere illusory projection from an observer’s mind. The thing is… math is absolute truth. Everything can be mathematically defined in some way. The Pythagorean Theorem will always hold for any right triangle on a perfectly flat surface, and methods of topology and the Law of Cosines can compensate for other distortions of triangles and other angles. It’s truly beautiful.

The traditional definition of chaos—where things run uncontrollably and rampant—is the antithesis of beauty. Mathematical chaos, on the other hand, is the epitome of beauty. Take the Mandelbrot Set, for example. It’s nothing more than taking a point on the Complex Plane, putting it into a function where one term is squared and added to the initial term, then fed back into the squared term, and looping—iterating it—for enough times that the point either goes out of bounds or doesn’t. Yet this simple generation does not generate a simple shape. In fact, it generates one of the most complex structures in all of mathematics—a fractal.

Due to the principles of Chaos Theory, with “time” being the number of iterations done to the point, a point that is a billionth of a billionth of a billionth of a billionth of a pixel away from the one right next to it might be in the Mandelbrot Set, while the neighboring point isn’t. The Mandelbrot Set is almost unanimously agreed to be the most be the most beautiful object in all of mathematics, as it is an extremely elegant demonstration of Chaos Theory.

Now think about Conway’s Game of Life. Imagine an infinitely huge sheet of graph paper. Each cell can either be shaded in or not shaded in—that is to say, alive or dead… and no, there are no Schrodinger cells. Each cell has eight neighbors. Let’s say we have a living cell. If there are one or no neighbors, the cell dies, as if of loneliness. If there are two or three cells, the cell lives on to the next generation. If there are any more neighbors, the cell dies, as if by overcrowding. If three neighbors surround a dead cell, the cell comes to life. From these simple rules, we can again use the principles of Chaos Theory to create a game board that essentially looks… living, hence its name.

To a layman, this may not seem all too fascinating. To a mathematician and mathematical biologist, these patterns take us one step closer to describing our own universe. Could we not be the byproduct of a very large array of an n-dimensional version of the Game of Life? Who is to say we aren’t? While this theory lies forever in the Conjecture Limbo of Gödel’s Incompleteness Theorem, it is certainly a sobering thought to think that our universe may be governed by a set of rules that an elementary school kid could understand.

Most people think of math as the horrible way schools teach it. I do not like school mathematics. I do not like it one bit. Why don’t I like school math? All of the classes—even Geometry—manages to squeeze out all of its elegance and beauty, and show us a framework that looks—let’s face it—ugly. It’s horrible—it’s like teaching that art is all about painting different types of fences. Math is an art, and its beautiful constituents—ESPECIALLY when it comes to geometry—are ripped out of the subject and presented to us in school. No wonder why kids do not like math. They don’t know what math is.

Math is an absolutely beautiful topic. Chaos Theory and fractal geometry are some of the many elegant, absolutely lovely mathematical topics. If I were to invent a math class, I’d teach all about these amazing concepts of math—what real math is, mixed in with some geometry. In an art class, people learn about the life of the artist. Why shan’t we say the same for math classes—let’s learn about, at least in part, the life of the mathematician. After all, math is none other than an art that uses symbols as metaphors to represent reality.

I may be only 17, but one does not have to be old to be wise. It simply takes a little elucidation to what really is to be captured by its allure, drawn towards not a Siren Song, but the calling of absolute truth and reality. Mathematics is not the greatest human invention. Mathematics has always existed and always will exist as absolute truth. Mathematics is the greatest human discovery. It may not seem like it at first glance, but it’s like a chocolate éclair—looks are deceiving, and all you need to do is to take one bite and you’ll have wondered why you ever hesitated to dig in.

I don’t know what else to add. It’s 6AM. I should get ready for school.

To say that we can’t quite solve the three-body problem is a common misconception. I recommend reading about the work of Karl Sundman.

http://rjlipton.wordpress.com/2014/07/25/an-old-galactic-result/ is a good intro to the topic; http://en.wikipedia.org/wiki/Three-body_problem#Sundman.27s_theorem has a bit of the actual result. “The global solution of the N-body problem” (Wang Qiu-Dong) gives a generalization to higher orders.

It’s … complicated. Both the solution, and the nature of the solution — that it converges but is impractical for almost all applications. Alas!

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Thank you for the links! I shall read them to spend my life being a little less wrong!

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