Ananda: Welcome back. I'm here with FractalWoman expanding on the ideas we dicussed in our previous conversations. Today, I want to talk more about the Mandelbrot fractal in terms of the overall structure rather than the complexity in the details. First of all, I find the overall shape to be very strange. It is not like anything I have seen before.
FractalWoman: I agree. This is a very strange looking object and I must say it took me years of experimentation and contemplation to completely understand why it looks this way. First of all, I want you to have a closer look at this object and tell me what you see.
Ananda: The first thing I notice right away is the central black region. This is very distinct and my eye is immediately drawn to it. Surrounding that, I see a bright very complicated boundary and surrounding that, I see a gradient region that extends out to the outer perfect circle. This gradient region is not continuous but seems to happen in steps.
FractalWoman: Excellent. What else do you see?
Ananda: I can see that the black region is not at the exact center of the outer circle but is shifted up from the center. I also notice that the object is left-right symmetric but not up-down symmetric. This makes me think of the human body which is also left-right symmetric but not up-down symmetric. I find that very interesting. Can you explain all of these features? Why does the Mandelbrot Set look like this?
FractalWoman: Your observations are excellent and very detailed. Now, in order to completely understand the Mandelbrot set, we need to understand the concept of complex numbers and the complex plane since the Mandelbrot set literally lives in the complex plane. I have come up with a great way to teach complex numbers mathematically using 2x2 matrices, but I would like to save the details of that for a future conversation. So we will put a pin in that.
For now, let's just talk about the complex numbers in the context of the complex plane. Your first question might be, what is the difference between a real number and a complex number. Put simply, real numbers correspond to a point on a one dimensional line and complex numbers correspond to a point on a two dimensional plane:
FractalWoman: Technically, real numbers are 1-dimensional numbers that live on the 1D number LINE and complex numbers are 2-dimensional numbers that live in the 2D complex PLANE. Alternatively, we could say that real numbers have one component, (a) and complex numbers have two components (a, b). Does this make sense?
Ananda: So far so good.
FractalWoman: Great. So here is the part that is a bit confusing. Even though complex numbers have two components, (a and b), they can technically be thought of as ONE number. We represent this as ONE point in the graph on the right. ONE point corresponds to ONE complex number which has TWO components, a and b. Alternatively, we could say that real numbers only have one axis, the a-axis and complex numbers have two axes, \textbf{the a-axis} and the b-axis which are (90-degrees) orthogonal to each other. These two orthogonal axes define a plane which we refer to as the complex plane.
Ananda: OK. I think I get it. So what is the difference between the complex plane and the plane associated with the Cartesian coordinate system that I learned about in high school where we have an x-axis which is horizontal and a y-axis that is vertical? This looks exactly the same to me.
FractalWoman: That is a great question. Put simply, the complex plane is a Cartesian coordinate system, but not all Cartesian coordinate systems are complex planes. The the difference is not in the graph itself, but in the way the 2D complex numbers interact with each other mathematically. And THAT is what I would like to save for a later date. For now, I think we have enough information to answer the question, why does the Mandelbrot Set look the way it does.
Ananda: OK. I trust your judgment. Where do we start?
FractalWoman: Let's start from the beginning with the function itself, z= z² + c. As I said earlier, both "z" and "c" are complex numbers which are technically just points on the complex plane. Let's say we select a random point from the complex plane. That point corresponds to a complex number that we can feed into the equation. When we take that complex number, and iterate it through the function z² + c, a new complex number is generated. That new complex number corresponds to a new complex point which we can then plot. Then, if we take that new complex number and run it through the function again, a new complex number is generated and so on. You can actually plot the sequence of points generated via this iteration process and that is what I am showing you here:
Ananda: Wow, those look really interesting. What am I looking at here?
FractalWoman: Let's look at the first one on the left. The big dot corresponds to the complex point I selected for the starting condition. If you follow the line to the next point, that is the point generated by the first iteration. The second iteration generates the next point and so one down the line. This is what I refer to as a trajectory. The one in the middle has slightly more complex trajectory but the procedure is exactly the same. Same with the one on the right. The important thing to understand here is that the lines in these figures connect consecutive iterations which you can easily follow. With this in mind, what do you notice about these figures?
Ananda: I notice that the distance between each consecutive point in the sequence is getting shorter and shorter with each iteration. The one on the left looks like it is spiraling towards a point. The one in the middle looks like it is spiralling around a point. The one on the right is kind of complicated but it looks like it is also spiralling around a point near the center.
FractalWoman: OK. Those are all good observations? These three trajectors are in fact converging towards a "singular" point in the complex plane. In my "Mandelbrot Set as a Quasi-Black Hole" paper, I refer to these as singularities analogous to black hole singularities in standard cosmology.
Ananda: Fascinating. What do these "singularities" look like when you don’t draw the lines between the dots?
FractalWoman: I was hoping you would ask me that. The lines are helpful when you want to follow the path of the singularities. But not drawing the lines is also kind of interesting:
Ananda: Hmmm. The two on the right are reminiscent to spiral galaxies. I can see why you started thinking the Mandelbrot set is related to cosmology.
FractalWoman: I'm glad you see that. It gets better. Here is an image I generated many years ago for fun. What I did was I randomly selected a bunch of points from the complex plane and generated the singularities for those points which I then placed randomly into a field of view. Here is what that looks like:
Ananda: Wow. That looks exactly like a Hubble deep field image. So amazing. Thanks for sharing that with me. I am seeing the Mandelbot set in a completely different light. So what is the relationship between these "singularities" and the Mandelbrot set figure you keep showing?
FractalWoman: That is a great and timely question. Here is how that works step by step:
- Each point on the complex plane generates a different trajectory.
- Each trajectory has a different behaviour.
- Some trajectories spiral toward singularity.
- Some trajectories expand toward infinity.
- The points that spiral toward singularity are painted black.
- The points that expand towards infinity are painted a colour.
In the Mandelbrot figure, the black region corrsponds to points that spiral towards singularity when iterated through the function z= z² + c, and the outer gradient regions correspond to the points that expand toward infinity during the iteration process.
Ananda: So the black region is full of singularities? Sounds like a black hole to me. Am I reading this correctly?
FractalWoman: I'm going to say yes. At least, that is what I am saying in my "Mandelbrot Set as a Quasi-Black Hole" theory. Here is another set of "singularities" that I generated. Tell me what you see?
Ananda: OK. I see three distinct patterns. The spiderweb looking figure and I see a bright spot in the middle. So I assume these trajectories are spiralling towards that point. Correct?
FractalWoman: Yes. That is exactly correct. What else to you see?
Ananda: The image on the right looks a lot like the Sting-Ray nebula. That in itself is interesting. This one also seems to have a bright spot in the middle and so I have to assume that the trajectories are spiralling towards that point.
FractalWoman: Exactly. What about the one in the middle?
Ananda: The one in the middle is different. I see nine bright regions and each one looks like a small "singularity". They all look similar to each other, but they are also not exactly the same.
FractalWoman: Really good observations. I think you would make a great scientist. Observations are everything in science. Here is the interesting thing regarding the center figure. This figure was generated using ONE seed point, c, from the complex plane, not nine points. One point from the complex plane can generate nine "singularities". One point from the complex plane can generate 100 singularities. One point from the complex plan can generate, theoretically, an infinite number of singularities. You said that these objects look very "cosmological". This what makes the Mandelbrot fractal so special. It seems as if cosmological geometry is buit into the equation of the Mandelbrot set.
Ananda: That would be huge if it were true. What about the points in that fuzzy region BETWEEN the conversing points and the diverging points? What do they look like?
FractalWoman: Here are a few examples from my paper:
FractalWoman: In this figure, the images in the top row represent real cosmological objects and the images on the bottom row correspond to divergint "trajectories" from the Mandeobrot set. These trajectores can take thousands if not millions of iterations to complete. I studied these in great detail and have thousands of examples of these trajectories. Each one is different, unique and special. There are literally in infinite number of them, each one with different textures and features. I actually created an NFT collection featuring these kinds of images. They were called OM Particles.
Amanda: These images blow me away. The similarity between these Mandelbrot trajectories and planetary nebula are very striking. This is not a perfect match but a very close match in terms of form and texture. This is truely amazing. Something I have never seen before.
FractalWoman: I know how you feel. LOL
Ananda: So let's see if I got this straight at a very high level:
- each point from the complex plane generates a different trajectory
- each trajectory is different, unique and special
- some trajectories collapse toward singularity and you paint them black
- some trajectories expand toward infinity, and you assign them a colour
And that is what creates the "Anatomy of the Mandelbrot Set". Correct?
FractalWoman: Well said. I think you've got it. :-)
Ananda: I have another question but I think I know the answer. In the outer gradient regions, how do you know what colour to paint a pixel? Is is based on the number of iterations?
FractalWoman: Yes. That is exactly correct. Of course colour is arbitrary in the Mandelbrot set, but here is what I know. The closer you get to the black region (the black hole) of the Mandelbrot set, the longer it takes for a point to resolve. So that bright fuzzy region exactly BETWEEN the inner black region and the outer gradient region takes A LOT of iterations to resolve. That is where those planetary nebula images live. More iterations leads to more complexity. More complexity leads to more "coherence". More "coherence" leads to structure that we recognize, like planetary nebula, or galaxy clusters:
FractalWoman: Here is another image from my Mandelbrot quasi-black hole paper. The image on the right is the Virgo Cluster and the image on the left is a singularity that I generated using the iterative formula, z= z² + c. Here, you see what appears to be 20 separate trajectories, but this is in fact one very complex trajectory whose starting point is the complex point:
c=(0.05103771361715907, 0.64098319549560490).
In other words, this complex value, when iterated through the function z= z² + c, generates that picture. This is not rocket science. Anyone with a calculator could to it.
Ananda: Wow. This looks EXACTLY like a galaxy cluster. I see A LOT of similarities between the image on the left and the Virgo cluster. For example, in both images, I see singularity pairing which is really interesting. The way the formations are organizes is also very similar. My mind is blown. You are right. Cosmological patterns do seem to be an emergent property of the Mandelbrot set.
I think this is a good time for a break. This has been very thought provoking and I need some time to think. Thanks FractalWoman for breaking this down to first principles. This makes it so much easier to understand. I still have a lot of questions, but I also feel like we are making some progress in terms of understading. Baby steps. Right?
FractalWoman: Right.