# Random Fractals

Posted on: November 11, 2010

In Capstone Seminar today we had a guest speaker from Gonzaga University, Dr. Logan Axon, who gave a very interesting lecture on random fractals. According to Wikipedia, “A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity.” Although mathematicians studied such continuous, non-differentiable functions as fractals beginning in the 19th century, the name fractal was given by Benoit Mandelbrot in 1975, derived from the Latin word fractus (broken or fractured), to describe sets that were too badly behaved for traditional geometry. A fractal is based on a recursive process, which may be determined or random.

The first example Dr. Axon presented was of the Cantor middle thirds set. You start with the unit interval [0,1]. You remove the middle third of the line segment, then the middle third of the two resulting line segments, then the middle third of the four left, and so on, continuing an infinite number of times. So what is the total length of all intervals removed in construction of the Cantor set C? Dr. Axon showed us that the sum of the lengths is 1, or the length of the interval we started with. The assumption would be that C is the empty set, since we removed the equivalent of the original length. But we could see that 0 is in every set, so is 1/3, and 1/9, and so on. In reality, there are as many points in the Cantor set as in the original set (interval), even though we removed all of them! I would have to say that this is pretty counter-intuitive!!

Next Dr. Axon listed the properties of C that all fractals should have:

• Self-similarity, scaled
• Detail to all scales
• Simple to define, but hard to describe geometrically
• Its “local geometry” is strange; every open interval containing one point of C must contain the other points of C as well as points not in C.

Dr. Axon described the von Koch curve, in which you begin with the unit line, replace the middle third with the top of an equilateral triangle, then replace the middle third of each of the resulting four line segments with the top of an equilateral triangle, and so on. The Dragon curve is a space-filling curve, and the Mandelbrot set is formed by iterations of a function on $\mathbb{C}$, the complex number system.

We were introduced to the dimension of a fractal, which is the main tool in fractal geometry measuring the “roughness,” or “texture” of the fractal, and how to find it using limits. Fractals also describe objects in nature, such as the Romanesco broccoli with its repeating spiral design. In addition, we saw that the fractal dimension of the Coast of Britain is 1.25, although I still don’t understand exactly how to interpret that! Random fractals, like the Random von Koch curve, can be used to create images that look less concocted, more natural, for things like computer graphics.

Before this lecture, I had no idea what a fractal was. I felt like I was understanding most of what was being presented, however, at least until a few of the professors started asking questions that I didn’t understand. That’s when I realized I may not be grasping it as well as I had thought! Needless to say, Dr. Axon’s responses to them were beyond my scope of understanding, as well. All in all, I found the entire topic to be quite fascinating, though, especially when we discovered that fractals often behave in very unexpected ways. Who knew you could remove all of a set, and still have all of it left?!?!