Ananda: Welcome back. I'm Ananda, and I'm here with FractalWoman, hanging on for dear life as she walks us through a paradigm shift in thinking. Previously, we talked about how the standard model of cosmology might be missing something, and how the fractal paradigm could offer a radically different, maybe even more intuitive, view of the universe. Today, we'll be doing a deep dive into the Mandelbrot Set itself and how it is generated. I personally would really like to understand this better.
FractalWoman Ananda, I just want to say that you are taking all of this very well. You are doing better than "hanging on for dear life". Truthfully, you seem to be getting it better than most. I am 100 percent sure you will be able to understand what I am about to teach you. So where were we. Oh yes. As I said before, the fractal cosmology that I am proposing puts the Mandelbrot fractal at the center of the cosmological stage. Given this, I think it is important to study all aspects of the creation process of this particular fractal. It may seem complicated at first, but once we piece it all together, you will start to see how simple it really is.
Ananda: I hope so. I always thought of the Mandelbrot Set as mathematical art, but I'm starting to see that there may be more to this "mathematical monster" than meets the eye.
FractalWoman: You're not alone in that. Most people that encounter the Mandelbrot Set see it as mathematical anomaly and nothing more. But under the hood, there’s something much deeper going on. What you’re looking at is the output of a recursive function that repeats a simple rule z → z² + c.
Ananda: That’s still hard to wrap my head around. How does that such a simple formula produce such an infinitely complex structure?
FractalWoman: That’s the magic of iteration. Let's have a look at the first five iterations then I want you to tell me what you see:
Ananda: First of all, I see that the function is getting longer and longer with each iteration. So the function itself is becoming more complex after each iteration.
FractalWoman: Excellent. What else do you see?
Ananda: Hmmm. I notice the original function is squared, then the result gets squared again, and then that result gets squared again and so on. So each time we iterate, the overall power of the function keeps growing.
FractalWoman: Excellent observation. In fact, the degree doesn’t just grow, it doubles with every iteration. After one step the highest power is 2, after two steps it becomes 4, then 8, then 16, and so on. The complexity expands exponentially as the iteration continues. What else do you see?
Ananda: I see that the original formula, (z² + c) is still in there. It never goes away.
FractalWoman: Wow. Very impressive. I wasn't sure if you would get that one. Yes, all of the previous functions are stored in the higher order function that is created with each iteration. In other words, the iteration process does not destroy information. It merely stores it and reuses it. This is an important point to keep in the back of our minds for future discussions. It will come up again.
Ananda: So now I can see how complexity can come from simplicity. Each iteration generates a higher order, more complicated function than the previous iteration. Here, we only did five iterations and the function already looks more complicated than anything I have encountered before. I can only imagine what the function would look like after 100 iterations or 1000 iterations. What if we did 1 million iterations? This makes E = mc² look like child's play.
FractalWoman: I agree. I think we are making great progress here. This totally explains the vast complexity of the Mandelbrot fractal. Of course all iterative functions behave this way. The reason the Mandelbrot Set is special is because it is one of the simplest iterative functions that exists in mathematics and yet it is maximally complex in the patterns that it can generate. This is "complexity from simplicity" at its best.
Ananda: It really is amazing. But I feel like I’m still missing something important. I can see how the function grows with iteration, but how does that actually create the picture of the Mandelbrot Set? How do we go from an expanding equation to that famous fractal pattern?
FractalWoman: That is exactly the right question to ask. Up until now, we have been looking at how the equation grows internally and how each iteration builds more algebraic complexity. But the Mandelbrot fractal is not defined by how long or complicated the function becomes. It is defined by how the values produced by the function behave as the iterations unfold.
Here's how it works from a very high level. First we select a value from the complex plane and call it c. Next, we set z=c. Then we recursively apply the rule z → z² + c and we watch what happens after each iteration.
Ananda: Wait a minute. Sorry to interrupt, but I thought z was suppose to start at zero?
FractalWoman: Technically, that is true, but when we start at z=0, then the first iteration always sets z=c, so we might as well start at z=c. At least that is they way I do it. I can't think of any good reason to start at z=0 when the first iteration is always going to set z=c. As a computer scientist, I am always looking for ways to make my programs more efficient and setting z=c reduces the number of calculations I have to do.
Ananda: Oh, I see. Thanks for clearing that up.
FractalWoman: No problem. Every detail is important so please feel free to stop me any time you have a burning question.
Let's go over this again. First we select a value from the complex plane and call it c. Next, we set z=c. Then we recursively apply the rule z → z² + c and we watch what happens during the iteration process.
What we found is that each different complex value, c, behaves differently when iterated through the function z → z² + c. To generate the above picture, what we do is we map these different behaviours to different colours. For example, the points in the central black region behave one way during iteration (they collapse and converge), and the points in the outer gradient regions behave in a completely different way (they expand and diverge). This is what gives rise to the two distinct regions of the Mandelbrot Set. Does this make sense?
Ananda: Yes. I think so. Different complex values lead to different behaviours. I find that very interesting. I would like to find out more about these different "behaviours". What do they do and more importantly, what do they look like?
FractalWoman: For sure. But maybe we should save this for next time. What I can say is that you are going to be very surprised when you see "what they look like".
Ananda: Hmmm. Sounds interesting. Can't wait.