| 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) |