Ananda: So here is my first and most obvious question. I hear the word “fractal” gets thrown around a lot these days. But what exactly IS a fractal?
FractalWoman: What does it actually mean to be a fractal? That is a good first question. Put simply, a fractal is a pattern or structure that repeats itself at different scales. In other words, when you zoom in on a fractal, you will see shapes and patterns that look similar to each other and similar to the original structure. This is what we refer to as self-similarity. This property of self-similarity appears to be everywhere in nature, from tree branching and meandering rivers, to your vascular system and the structure of the lungs in our body.
Interestingly, both trees and lungs have evolved to serve a similar (self-similar)respiratory function in nature. So you can see how important this fractal feature is in terms of natural systems and the existence and persistence of life on planet Earth. Long story short, the property of self-similarity is the key signature of a fractal and that is what we look for when we are looking for fractals in nature.
Ananda: So it’s kind of like a pattern that echoes itself?
FractalWoman: Yes. That is an interesting way of looking at it. And it’s not just about the pretty patterns that they form. Fractals are the basis for a mathematical language which can be used to describe systems that are complex, irregular, and dynamic. Benoit Mandelbrot, who coined the term "fractal," once said that clouds are not spheres and mountains are not cones. Nature isn’t smooth. It is rough and wrinkled and layered, and fractals help us describe these kinds of patterns. We refer to this as "fractal geometry".
Ananda: What about lightening and coastlines? They are not ordered regular shapes, but they’re not completely random either. In both cases, there appears to be a recognizable pattern. For example, I can tell the difference between lightning and a coastline just by the shape and texture of the patterns.
FractalWoman: Yes, it is true that lightning and coastlines are fractals since they have the property of self-similarity. If you can zoom into a part of a structure, and see something that looks very similar to the original structure, then you are likely dealing with a fractal.
Ananda: Many fractals, like the coastline and lightening are still kind of predictable once you get to know the pattern. But some fractals, like the Mandelbrot set (and the Julia set), are extremely complicated with a lot of variations in shapes and textures. And when you colourize them, they become even more incredible.
I played around with your fractal flythrough software and found that I could zoom in seemingly forever with NEW patterns appearing as I zoomed in. I find this extremely fascinating. I never get tired of flying through this fractal because I never know what I am going to find next.
FractalWoman: Great. I'm glad you enjoy the experience of flying through a fractal as much as I do. I also never get tired of it and I have been working with the Mandelbrot Set since the mid 1980's. It is one of those things that never gets old.
Ananda: I was even more surprised when you showed me the "equation" that you used to generate these images and I saw how simple it was. It is my understanding that the computer code for this is also very simple. How does this work? How can this incredible complexity come from such a simple formula?
FractalWoman: That is a great question. In the fractal paradigm, complexity can emerge from simplicity via some sort of iterative feedback loop. That is the key. Without a feedback process, fractals could not exist. There are many systems like this where you run a set of simple recursive instructions or rules and it evolves into something wildly intricate. But the Mandelbrot set is the most interesting one in my opinion.
The Mandelbrot Set
Here is a simple graphic I made that shows the main features of the Mandelbrot Set. As you know, the formula to generate the Mandelbrot Set IS extremely simple:
z = z² + c
Ananda: Yes, almost as simple as E = mc².
FractalWoman: For sure. But here is the difference. In the equation (E = mc²) all of the values in the equation (E, m and c) are real numbers. However, in the Mandelbrot formula, both z and c are complex numbers. Complex numbers are not the same as real numbers, but we are not going to talk about that...yet. Suffice it to say the complex number system is an extension of the real number system. They are different, but not as different as you might think.
More importantly, you will notice that, in the Mandelbrot formula, "z" is on both sides of the "equation". This is indicative of a recursive function where the output of the function is fed back into the function forming an iterative feedback loop. In fact, this is not an "equation" at all, it is an iterative function and is better written like this:
z → z² + c
This reads "Z goes to Z squared plus C" or "Z becomes Z squared plus C".
This recursive iterative process is the key to understanding how fractals come into being. THAT...is how complexity can come from simplicity.
Ananda: This makes me think of what happens when you place a microphone in front of a speaker and get all that feedback noise. If we could look at this feedback noise, would it look like a fractal?
FractalWoman: Yes, absolutely. You are thinking in the right direction. The feedback noise (like when a microphone is placed too close to a speaker) can exhibit fractal characteristics, especially in how the sound waves evolve over time. The noise often contains bursts of echoes within echoes, creating nested self-similar structures which of course, is the hallmark of a fractal.
Ananda: Wow. Do you have any idea what that would look like if you could see it?
FractalWoman: I’m not sure what audio feedback looks like, but I do know what video feedback looks like. I played around with this back in the day when we had analog video cameras and CRT television screens. All you have to do is plug the camera into the TV and then point the camera at the TV. When you do that, patterns like this begin to appear on the screen:
Ananda: That is really cool. They look very mandala like. I really love the Mandelbrot fractal. It is my favourite fractal by far. As I was playing around with the software you gave me, I couldn't help but wonder: does the Mandelbrot fractal ever repeat itself?
FractalWoman: Not exactly. As you zoom in, you may find SIMILAR patterns (self-similar), but the patterns are never exactly the same. That is the amazing thing about the Mandelbrot set. You can think of it as a never ending pattern generator.
Ananda: Is the Mandelbrot fractal infinite?
FractalWoman: Theoretically, the Mandelbrot fractal does contain infinite complexity. It is only limited by the digits of precision of the computer that is generating it. So the computer itself introduces limits to the complexity. Theoretically, if you could make a perfect computer with infinite precision, you could zoom in forever generating an infinite number of complex patterns, each one different (but similar) to all the other patterns.
Ananda: Interesting. In the Mandelbrot figure, I see a fuzzy curvy pattern just outside the black region. Is that were all the fractals are found?
FractalWoman: Yes.
Ananda: I also notice that this fuzzy pattern is completely contained within the outer circle shown in this figure. So based on what you said, it appears that infinite complexity can exist within a finite "space", theoretically. Right?
FractalWoman: Exactly. That is a great observation. In the fractal paradigm, infinite complexity can be contained within a finite boundary condition. Theoretically of course. This, in my opinion, explains how a finite universe can contain infinite complexity. In other words, the universe can be both finite and infinite. Using standard geometry, this is a contradiction, but using fractal geometry, it is completely doable. We could in fact be living in a fractal universe that is both finite and infinite, not unlike the Mandelbrot fractal.
Ananda: Wow! You are blowing my mind. So you’re not just saying there are fractals in the universe. You’re saying the universe is a fractal?
FractalWoman: Exactly. That is the core idea of the fractal cosmology that I envision. The idea is that we are embedded in a cosmic fractal, and the creation and evolution of the universe (and everything we observe and perceive) is an emergent property of some sort of universal iterative feedback process. Not only do I believe the universe is a fractal, but I believe the universal fractal is very closely related to the Mandelbrot fractal.
Ananda: That’s a wild idea. Is there any evidence of this?
FractalWoman: Well, I have been collecting evidence of this for many years now. Here is a figure from a paper I wrote called "The Mandelbrot Set as a Quasi-Black Hole" that I like to use when I get asked this question. The image on the left is from the Mandelbrot set and the image on the right is of the Grand Spiral Galaxy, NGC 1232:
The similarities between these two structures is striking in my opinion. There are many more examples of this in my paper. I might even be inclined to say that the universe is self-similar to the Mandelbrot fractal. Not exactly the same, but very similar. A fractal worldview changes how we see our place in the universe. We are not separate from it. We are an emergent property of The Universal Fractal.
Ananda: Wow. The similarities between these two images surprisingly obvious. The spiral patterns are almost exact. The way the information clumps is also VERY similar. One might be inclined to say "self-similar". If I were a scientist, I would call this GOOD evidence.
I think it's time to take a break. Fractal Woman has left us with a lot to think about. Don't worry. We'll be back.
References:
Barnsley, Michael F. Fractals everywhere. Academic press, 2014.
Gardi, Lori. "The Mandelbrot set as a Quasi-Black hole." Proceedings of CNPS (2017).