Fundamental Equations, Chaos, Fractals, and Leaky Faucets

It’s been a while since I’ve written here. Busy teaching. (I know, lame excuse).

So I go and watch some video on YouTube, and on the side is the list of suggested videos. For a while there’s been this one by Veritasium (Derek Muller). Now, Derek is a fantastic science educator, and is who I want to be when I grow up. One problem is he’s younger than me by over a decade. Hmm. Have to work on that somehow.

Anyway, the video that was just waiting for me to finally click on it was this one. It’s about an equation that will change how you see the world.

The Logistic Map

The math is very simple: x_{n+1}=r x_n(1-x_n) where r is the growth rate. This is a very simple equation with a negative feedback loop.

When you graph r by the equilibrium population, you get this:

from the Wikipedia article on the Logistic Map


Once the growth rate hits 3, the equilibrium population splits, and oscillates between two values. Then just after 3.4 it splits again. And very soon it becomes chaotic. Oh, and fractal. The chaotic nature was used for pseudorandom number generators.

The Mandelbrot set

Does mentioning fractals make you think of the Mandelbrot set? If it doesn’t, then you have some research to do.

From the Wikipedia article on the Mandelbrot Set

It’s probably the most famous fractal out there. Heck Johnathan Coulton has done a song about it.

But evidently if you somehow rotate the Mandelbrot Set along it’s real number axis, you get this:

From Physics Forum

Look familiar? At this point I started getting a headache, but it was one of those good, excited headaches that come from having your reality twisted about.

Leaky Faucets

Oh yeah, leaks. Derek then mentions that if you get your faucet going drip, drip, drip, and then increase the water pressure just right, it will start doubling: drip drip, drip drip, drip drip. Push it a little more, and you get chaotic behavior.

Of course the YouTube video is so much better than my explanation. Go watch it.