Mandelbrot's Evil Twin

Video Description

MUSIC BY 6884! Check him out here: https://6884.bandcamp.com/ The soundtrack from this video will be available on his bandcamp soon, so go follow him to make sure you don’t miss it! I know I promised you a lambda calculus video- In fact, the video is done, but 6884 is having a field day on the sound effects as we speak, and they are coming out just as sick as this video did. Don't worry, your half-hour saga of animated tromp diagram beta reductions is on its way :D Rendered with SwapTube! https://github.com/2swap/swaptube Technical deets for the nerds: First of all, I am using a simple escape-time algorithm with a bailout at radius 256. I understand that this approach is, in some sense, inappropriate at revealing structure in mandelbrot sets for negative real exponents, since divergence towards infinity is not expected. I have no reason to suspect it would behave either for, as example, complex exponents with negative real part. If you pay attention during the X-Set tour, you can generally see a difference in behavior on the left and right side of the screen, and I figure this is the reason, since the left side corresponds to negative real values. There are alternative strategies for dealing with things like this, such as using the Lyapunov exponent to check for chaotic or periodic behavior, but I opted to go with the simpler and more familiar approach to rendering. After all, I didn't want to taint the positive-real-part results with an approach designed for the negatives. Another fun little quirk that I ran into here- the standard approach to anti-aliasing the color involves determining the distance that a point gets from the bailout radius after achieving bailout. This method requires knowledge of exponent- you expect greater "jumps" as the exponent gets bigger. On the flipside though, as the exponent approaches 1 from above, this approach produces noisy behavior (and below 1 it simply doesn't work.) So, if you pay close attention, you will notice I smoothly transition between gradation and non-gradation up to the parameterized value which I'm plotting in the video :) I am also wondering what's up with the stripey behavior for some cross-sections of the X-Set. I unfortunately have not gotten around to investigating, but my best guess is that it has something to do with branch cuts on the complex natural logarithm (as used to compute complex exponentiation.) You know the left-pointing "needle" of the mandelbrot set? During all my transitions between the X-Set and Mandelbrot set, for example at 00:42, try and find the shape in the X-Set that transforms into the needle. Notice anything interesting? :) If you have any references about this set, let me know!