Julia Set Animations from 2023

I made these when I was a math teacher. Hopefully you think they are cool.

Check out the code in Binder (interactive) or on GitHub. The code is a Jupyter Notebook that contains a detailed explanation of how it works and how to use it to generate your own Julia/Mandelbrot renderings and animations. The code could be easily altered to make mandelbrot zooms

If you're unfamiliar, a Julia Set in this context is all the points on the complex plane that don't zoom off to infinity under the iterated mapping z->z^2+k. Each frame of these animations was generated with a different value of k.

100 Julia Sets with k near -0.8+0.156i following a circle with r=0.05 and center 0.8+0.156i

100 Julia Sets, k(t) = 0.05exp(it)-0.4+0.6i

Julia Sets are related to the more famous Mandelbrot Set. The mandelbrot set is generated in a similar way to julia sets, but instead of choosing one value of k for the entire complex plane, we use the mapping z->z^2+z_0, where z_0 is the initial value of z, before any iterations. My code can also make pictures of the mandelbrot set:

The mandelbrot set has many named features. Check out jlmarguz's Navigating the Mandelbrot Set for more info. This next gif of julia sets was created by choosing values of k near the "Period 2 Continent" of the Mandelbrot Set. This is the large circle to the left of the main cardioid, and it's called that because points in this circle end up in a stable cycle between two points under the mandelbrot iterated mapping. Julia sets are usually more "interesting" near the boundary of the mandelbrot set. and sometimes they look similar to what the mandelbrot set looks like in that area.

100 Julia Sets near the period 2 continent of the Mandelbrot Set

k(t)=0.26exp(it)-1.01

similar to the previous gif, but slightly shifted k(t)=0.26exp(it)-1

This one chooses k based on points close to the main cardiod of the mandelbrot set.