Weekly 7: Fractals

In class we did an overview of fractals, learning about Julia sets, the Mandelbrot set, and Sierpinski's Carpet. What these have in common is their repetitive nature: at every magnification, these sets have a repeating pattern that is a scaled copy of the original, fundamental pattern. That is, the patterns are self-similar and have an infinite degree of complexity. We saw this in Sierpinski's Carpet, and Sierpinski's Triangle is given below as another example of their construction.

Of course, at this step, there is not an infinite degree of complexity in the triangle. However, this pattern could be repeated as many times as you'd like, allowing for a level of detail that gets arbitrarily large. For a good visualization of the Mandelbrot set's complexity through magnification, watch this video. As is seen in the video, the original shape of the Mandelbrot set is seen repeatedly throughout the magnification (along with some other really interesting designs), showing us right away that there is no "maximum" magnification that can be achieved. Due to this property of being able to refine the detail of fractals as much as needed, they have become incredibly useful in a variety of situations. One of their uses is in graphic design, where fractals can be used to create a number of visual effects like the mountains seen below.

Using the same repeating patterns, programmers can recreate the appearance of natural phenomena with great detail. The method to create the mountains above can result in scenes that look incredibly realistic, which is quite a feat given the relatively simple nature of the mathematics involved in fractals.

As an example of the simplicity of the mathematics involved, let's consider the Mandelbrot set. It is created in the complex plane with the iterative equation

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A complex number z is chosen as the first value and entered into the equation. The result is then re-entered into the equation as the input value, and this process is repeated as long as wanted or needed. The visualization comes from assigning colors to values whose iterations either converge to a certain value or diverge. The familiar shape of the plotted set is achieved by testing each value c for convergence or divergence when the initial choice of z is 0 and respectively coloring the points.

In addition to their use in creating images, fractals have become beneficial to other areas of study. A telecommunications company, Fractenna, has designed a fractal-shaped antenna that is smaller and weighs less than the common antenna. Antennas are typically tuned for a specific frequency or range of frequencies, and the repeating patterns of fractals allow companies to reduce the size of the antenna required to serve these frequencies.

Fractals also appear in nature, from molecular structure to the shapes of galaxies. For example, we may look at a tree and notice its self-similarity: the primary limbs branch off from the trunk while smaller, secondary limbs branch off the primary limbs, and so on. In other words, each branch is a "trunk" for yet smaller branches that eventually terminate in leaves. A similar description applies to vascular and neural networks.

We worked on creating fractals of our own in class by hand and on the computer, and below is what I created using the site listed on our course page.

There were a lot of cool things you could do with the program, making different shapes that repeated themselves in different ways, and combining different patterns together to make increasingly complicated images. One thing that I noticed while playing with the program was how truly different an image can become given a small change to the initial pattern, which attests to the remarks made in class about chaos theory and the importance of initial conditions, and the vast changes apparently minor differences can make. Due to this effect, the image above took several tries to produce. Even once I was satisfied with the initial "stem" it was difficult to orient each sub-stem so that it gave the image I was trying to create because any error is magnified by the infinitely-repeated pattern I used as my base design. With this in mind, it becomes clear why it's difficult to accurately predict phenomena like weather and stock market prices with a high degree of accuracy, even if sophisticated models and technology are used.