Sounds of the Mandelbrot Set

[Music] it's no secret i like fractals well visually at least and in some cases as video games but what do fractals sound like i mean it's pretty obvious they taste good but listening to them the mandelbrot set an instrument no patrick the mandelbrot said this is not an instrument or is it so real quick i know this isn't a hyperbolica devlog this was just an idea i had and i was just so curious about it i had to try it so i coded up a quick prototype in just a couple hours now in order to explain the instrument i'm about to show you i need to first quickly explain what the mandelbrot set really is most people know it as a following you take any point c on the complex plane make a copy of it call it z and then just keep updating z with this equation thousands of times if the point eventually drifts off to infinity it's not part of the set and gets assigned a color but if it converges or goes into a cycle or something then it is part of the set and we color it black note that you don't actually need to use or even understand complex numbers regular old x and y work too it just makes the equation look less elegant but what is the path each point takes as it iterates well here i can actually show how that looks depending on which point you start with it will either converge to a point escape to infinity or converge to a cycle called an orbit these orbits can have different periods like three or four or five and different shapes by just choosing different points and so all this interesting structure and nuance is lost when we just color it black and it got me thinking what if we treat these orbits as sound waves then i could listen to different parts of the mandelbrot set and hear how they sound so the way this works is basically i just convert the x and y coordinates to the amplitude of the left and right speakers i choose a low sampling rate like 8 kilohertz and use some interpolation to smooth it to a more standard 48 kilohertz now it's really easy to tell the period of the orbit because you can hear the fundamental frequencies another neat thing is that as you zoom in each bulb adds another harmonic on top of the original one depending on which bulb you choose oh yeah and you can't zoom in too far because this all runs on the gpu in real time so that means it has a limited precision also these pure tones are kind of annoying so i'm going to add some dampening to make it more like an [Music] instrument [Music] although it's really fun to play around with the problem with the mandelbrot set is that all the interesting chaotic areas are unstable so you could never actually click on them because you'd need infinite precision so the orbit will always converge to some kind of repeating pattern which will just sound like a combination of pure tones if we want to hear some more interesting sounds we have to switch to some different fractals this one is called the burning ship fractal because that's exactly what it looks like over here the nice thing about this fractal is that it has chaotic regions that are actually stable and some sound really creepy [Music] [Applause] [Music] it's still a little hard to tell what's going on when the fractal is entirely black so i'll add some coloring based on the orbit [Music] [Music] there's regions that converge to a point to a cycle and to chaos in different ways speaking of fractal variants here's one that i came up with that i call a feather fractal there's a few reasons i like it first of all there's a ton of good clusters everywhere that have different notes to play so it feels much more like some kind of infinite piano where you can zoom in and find all sorts of different notes with different tonal relations [Music] the other reason i like it is that it just looks so cool i mean look how beautiful this is [Music] now let's switch from beautiful to ugly again this fractal i'm calling the sound effects fractal it doesn't look like much but it has an extreme variety of sound effects it can make because the orbits have a really interesting symmetry [Applause] looking at the orbit colors reveals a really rich and complicated structure which shows why there's so many sounds it can make it reminds me a lot of those sfxr programs that are used a lot in game jams and i'm pretty sure this fractal one could be useful for that too [Music] [Applause] [Music] if you're familiar with fractals already you may be wondering what about the julia sets well the reason i didn't mention them is because they're usually really boring at least in terms of audio for the mandelbrot set any point on the julia set will converge to exactly the same orbit as the point corresponding on the mandelbrot set now some other fractals can behave a little differently for example there's some julia sets of the burning ship fractal that are bistable a single julia set can converge to one of two orbits depending on the starting point and if you want to get even more exotic i found that some chaotic maps such as the chirokov map do actually have continuums for the julia orbits anyway this whole program is available on my itch.io page and the source code is on my github honestly it's easy to waste hours just exploring and finding new sounds so check it out and don't worry i'll be back with more hyperbolica stuff soon [Music] you

Channel: CodeParade

Views: 543,103

Rating: 4.9707627 out of 5

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Id: GiAj9WW1OfQ

Length: 9min 32sec (572 seconds)

Published: Sun Feb 28 2021

Reddit Comments

[u/Here_Man]

Great video

[u/bionicjoey]

CodeParade's videos mainly focus on the use of computer graphics to explore non-euclidan geometries, including fractals and hyperbolic/spherical spaces. In this video he shows off some tools he created to turn any point on the Mandelbrot set into a tone, then discusses how these tones and a visual representation of the gradients they follow can be used to gain a deeper understanding of the Mandelbrot set and similar fractal sets. Disclaimer: I'm not CodeParade, nor am I a professional mathematician.

[u/MammonStar]

sounds like a 56k modem

[u/JRGTheConlanger]

Undertone series