What are Fractals?
Interviewer: I guess the obvious questions to ask is “what are fractals?” and why are they important to they study of cosmology?
FractalWoman: Asking the question “what is a fractal?” is like asking the question, “what is a Universe?”. There is no easy answer. Although the idea behind fractals is quite simple, defining a fractal in words can be quite difficult, but I will do the best I can. Once you realize how easy it is to make a fractal, I am hoping you will see how easy it is to make a Universe as well. Personally, I am not a big fan of the current language of theoretical physics. It has become increasingly complex over the years, making it extremely difficult for the lay person (like myself) to understand the theories. The goal of my research is to simplify the language, making it both accessible and understandable to the lay person. I figured, if I can understand it, then anyone can. Do we live in a complex universe? Yes. The good new is, in a fractal universe, complexity comes from simplicity. This is the beauty of the fractal paradigm.
Interviewer: How can complexity come from simplicity? Isn’t that a bit of a contradiction?
FractalWoman: The Mandelbrot set fractal is a great example of complexity from simplicity. The formula used to generate this amazing fractal is (in my humble opinion) the most elegant formula ever written:
Z := Z2 + C
Not only is it absurdly simple, but the vast complexity that emerges from this simple formula is beyond imagination. I have been exploring this “mathematical monster” for many decades now and I always seem to find something new and remarkable about it. Interestingly, there are a lot of similarities between the patterns of the Mandelbrot fractal and the patterns that nature makes.
So I said to myself, maybe the Mandelbrot Set can teach us something new about how our universe works. In other words, maybe our universe is also governed by a simple set of mathematical rules that, over time, generates all the complexity that we see in the universe today. Maybe the laws of physics are emergent properties of fractal geometry and not the other way around. These ideas are in stark contrast to the conventional way of thinking which puts the laws of physics first and fractal geometry last.
Interviewer: So fractals are complex structures that emerge from some simple set of rules?
FractalWoman: That is one aspect of what fractals are, yes, but there is much more to it than that. Benoit Mandelbrot was the one who coined the term “fractal” to describe the rough or fragmented geometric shapes that we find in nature. Before the introduction of fractal geometry by Mandelbrot, the study of geometry was primarily limited to the study of simple shapes such as points and lines and simple curves. It turns out that the rough and complex shapes of nature cannot be easily described using standard geometry. Nature is clearly doing some “other” kind of geometry.
Interviewer: What is it about fractal geometry that makes it different from normal geometry?
FractalWoman: Self-similarity is one of the main feature of fractals that make them stand out from other geometric patterns. Structures with the property of self-similarity have similar “geometric” patterns appearing at drastically different scales. Below, we see a picture with the original Mandelbrot Set in the middle. Surrounding it are a number of smaller regions found within the Mandelbrot set that have a self-similar “black hole” region in the middle.
Self-similarity doesn’t mean that they are exactly the same. There are some fractals that are “self-same” like the Koch curve, but the Mandelbrot set is not one of them. Self-similarity means that there are some geometric features that are similar, but (not exactly the same) in different regions and more importantly, at different scales. This is how nature does it. Nature uses the similar patterns over and over again to create the complex structures we observe in nature. This is how the seed can be so small for instance. It only needs to store the information to make “one simple pattern”. This pattern can then be repeated over and over again at many scales to create intricate and robust structures. The seed growing into a plant is a great example of complexity from simplicity.
Interviewer In the above formula, I notice that Z is on both sides of the equation. This seems a bit strange to me. How can Z be equal to Z squared + C?
FractalWoman: This is actually just a problem with notation. Of course Z cannot be equal to Z squared + C and so, I understand why there might be some confusion here. What is happening here is that the output of the equation is being fed back into the equation in an iterative feedback loop. The Z on the righthand side is the input to the formula and the Z on the lefthand side is the output. You will notice that I used the notation := instead of = in the above equation. This is the notation used in computer science in a language called Pascal. The above formula loosely translates to “Z becomes Z squared + C”. The Z on the left is “new Z” and the Z on the right is “old Z”. The := notation is the notation that I use to indicate an iterative feedback loop.
All fractal patterns are formed by some sort of iterative feedback process. One of the very first fractal patterns ever generated using iterated function systems (IFS) led to an image that looks surprisingly like a fern. It is interesting to note that the fern is one of the oldest and most common plant species found on this planet, so it should not be too surprising that this pattern can be so easily replicated using the simple recursive rules associated with fractal geometry.
Interviewer: I know there are many fractal algorithms out there, many of which seem to be related the patterns of nature. Why do you think the Mandelbrot Set is special?
FractalWoman: The Mandelbrot set is a “universe” in it’s own right. It is theoretically infinite in that it is able to produce an infinite variety of patterns, and yet, it is completely bounded within the confines of a 2-dimensional plane called the complex plane. The iteration process is responsible for the generation of this complexity. Here is how it works.
Each time we iterate this function, Z := Z2 + C, more details are revealed. The image on the left is the first iteration. Beside it is the second iteration and beside that, is the third iteration. As you can see, with each iteration, the “curve” becomes more complex. The image on the right is what the “Mandelbrot Set” looks like after only 50 iterations. Each iteration generates more curvature. This is an important point. The curvature of a “fractal universe” ever increases over time. In standard cosmology, the “curvature” of space-time is said to be the cause of gravity. But how does “space-time” curve in the first place? In a fractal “universe”, curvature appears to come for free. If the universe is a fractal, then fractals like that Mandelbrot Set might be giving us a clue as to the origin of “space-time” curvature. This is why I think the Mandelbrot set is special. I think that it can teach us something about how the universe is generated and how it evolves over “time”.