It’s been suggested that ZRXC is not the “Real 3D Mandelbrot Set” because it fails to achieve certain visual standards.
This is a legitimate concern. I understand that to most people, the visual aesthetics of a fractal are the most important part. One of the goals of this essay was to show the reader something else, more important and beautiful beneath that.
Different people will have different standards for judging whether something is the “Real 3D Mandelbrot Set” — to me the Mandelbrot set is a step on the way to understanding a mystery, to solving a puzzle. So my generalization tried to fulfill that role better.
You are welcome to disagree, and I think a very strong argument can be made that the Mandelbulb set is more visually appealing than ZRXC… But I’d ask you make sure you understand the math outlined in this essay — if you skipped the sections on what the Julia Set and Mandelbrot set are, you missed the point of this essay (if you had trouble following, that’s to be expected since I’m not always the best at explaining things, please feel free to ask below).
Perhaps I should have titled this “the Natural 3D Mandelbrot Set.” Hindsight is always 20/20.
I think that is a good addition. If I may make one more suggestion, is that you state at the start of your article, how your goal differs from that of the Mandelbulb project: that you seek for mathematical elegance, while the Mandelbulb project first and foremost went for finding a 3D visual equivalent of the classic Mandelbrot shapes.
About skipping sections: the only bits that I skipped were the bits that I already knew :) I did love the visualisations of transformations in the complex plane, however. Very nicely done.
I did not understand what you were trying to say with the animated GIF about branches being arbitrary. Obviously something about the complex square root having two solutions, but how that relates to the image with rotating colours isn't quite obvious to me.
Also the part about the operations on sets isn't as clear as it could be. It's very intriguing though, yet another way of thinking of complex numbers that I didn't know yet :) I think it would become a bit clearer if you'd add axes to the image. Even better if you can also put angles (0, pi/2, pi, ..) in it.
BTW one very cool way in which complex numbers were explained to me when I was 16, does explain something you gloss over a bit in the "complex analysis" section. Why is the imaginary axis perpendicular to the real axis?
The way it was explained to me, is to think of the (real) number line, and how a multiplication by -1 is visually the same as a rotation by 180 degrees around the zero.
Now what if you'd decide to rotate 90 degrees instead? Such a crazy idea! You'd get a second number line. Let's call a rotation of 90 degrees to multiply by i.
So if we multiply 3 by i we get 3i, and if we multiply it again, it rotates 90 degrees further and we get -3. So that means 3 * i * i = -3 and i * i = -1, so i is the square root of minus one! Insanity! (the 16 year old me was giggling like a madman at this point)
(so yeah the explanation I got kind of started the other way around, with the 90 degrees angle first, and only "discovering" that this implied the square root of minus one after that)
Update
It’s been suggested that ZRXC is not the “Real 3D Mandelbrot Set” because it fails to achieve certain visual standards.
This is a legitimate concern. I understand that to most people, the visual aesthetics of a fractal are the most important part. One of the goals of this essay was to show the reader something else, more important and beautiful beneath that.
Different people will have different standards for judging whether something is the “Real 3D Mandelbrot Set” — to me the Mandelbrot set is a step on the way to understanding a mystery, to solving a puzzle. So my generalization tried to fulfill that role better.
You are welcome to disagree, and I think a very strong argument can be made that the Mandelbulb set is more visually appealing than ZRXC… But I’d ask you make sure you understand the math outlined in this essay — if you skipped the sections on what the Julia Set and Mandelbrot set are, you missed the point of this essay (if you had trouble following, that’s to be expected since I’m not always the best at explaining things, please feel free to ask below).
Perhaps I should have titled this “the Natural 3D Mandelbrot Set.” Hindsight is always 20/20.