Fractal Attractors

My journey into finding beauty in math.


Chaotic Attractors

Look, I didn't vibe with math in the past. I recall my SAT score for math was alright but I straight despised it. In adulthood I never worked jobs prior to tech that facilitated much math skills, even as a coffee roaster things didn't get much more complicated than fractions and division.
The Science Behind the Butterfly Effect is a video that Veritasium released in Dec 2019. Mr Veritasium aka Derek Muller goes on to describe the discovery of the first chaotic attractor by Edward Lorenz. A lot of big words are used and I can't even understand the jargon without spamming my tabs with wikipedia pages so here's the key point. Following Chaos theory, something is deemed chaotic if it exhibits sensitive dependence on initial conditions.

Chaos theory is a branch of mathematics that studies apparent random states of disorder and irregularities that are actually governed by deterministic laws that are highly sensitive to inital conditions.

I created these in p5.js, the javascript version of the processing library. These are specifically Clifford Pickover's equations for a chaotic attractor.
The exact equation is:

xn+1 = sin(a yn) + c cos(a xn)
yn+1 = sin(b xn) + d cos(b yn)
Where a,b,c,d are variables, 
that define each attractor.

Makes no sense right? Look I got all my math credits early. I dont know either.

How does one apply this equation?

Well this is a 4 parameter system, there is a numeric range that suits this equation best as far as beauty goes, being each variable should be in a range between -3 to 3. Within a graph each equation will provide a value to plot, one for the X-cord and one for the Y-cord.
Here's the p5 editor implementation of the Clifford attractor. Just click the play button to render it on your browser.
Be warned it can take about 30 seconds to render because It's plotting 1 million dots on a graph, it would take you a while too!