Trying to post a new fractal every day. Animations are coded with the help of https://teapaw.org/t.co/7pKEKo1gN7 by @ensl0
Joined March 2021
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This gives a little more context to yesterday's shader. I watched @3blue1brown's video on constructing a parameter space for the Newton fractal, and made this. The two roots are fixed at 0 and 1, and the remaining root is the parameter.
A closer look at what happens near the edge of those weird regions in the last animation. In this one, the stationary roots are at i and -i and the third root moves from sqrt(3) to about 1.8364. I also colored the roots white instead of black to avoid confusion.
Here's an example of a region where none of the points converge to a root. The orbits inside these regions "get trapped" and they don't escape, which is why they don't find roots.
These are also familiar shapes so it's kind of funny they show up here.
The two stationary roots are at 0 and 1, and the other root traces the dim path. You can find some neat patterns for when the roots are arranged this way, including regions where initial conditions fail to find roots.
We spent the last couple weeks or so looking at the family of Newton fractals in my complex dynamics class and @3blue1brown also talked about them over on youtube yesterday, so I wanted to share some of the stuff I wrote on smoothstep.io
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This fractal is actually the banner I use for this account. I coded it a while back on Shadertoy when @3blue1brown's Lockdown math series was coming out.
If you're interested, here's the video that sparked my curiosity: youtube.com/watch?v=elQVZLLi…
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Twitter compression hasn't liked these fractals recently so you should check out the code here for high quality: smoothstep.io/anim/f38bfae07…
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z->c*sin(z+k)
k = [0,2π]
Notice how changing k alternates between the sin and cosine fractal.
Also, here's the parameterization for k: desmos.com/calculator/xotbku…
It gives k some time to stabilize at intervals of π/2 to show the sin and cos fractals.
I'm back!!!!!
The big red dot is the initial point and the smaller red dots are the points after each iteration of the map. You can see what the mapping is actually doing by following the sequence of lines.
You can check it out for yourself here:
smoothstep.io/anim/adee1df5a…
In the last sprint of my semester, so I'm going to take a break from this account to focus on school for a week. I'll be back after my term ends👍
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