So what is the Mandelbrot set anyway? #fractal #tibees #math

I found that understanding the Mandelbrot set starts with understanding the behaviour of numbers when you square them. Some numbers when you square them blow up and go to infinity, but not all numbers. One says the same no matter how many times you square it, and numbers less than one actually get smaller when you square them. The Mandelbrot set explores this line between stable and unstable points, essentially take any complex number, square it, and then add it to itself. Take that result, square that, and add the same number on again. Do that over and over. If it takes you off to infinity, then your number is not in the Mandelbrot set. If doing that iterative process doesn't go off to infinity, but instead stays bounded, then your number is invited to be part of a special club and is inducted into the Mandelbrot set, which are the points shown in black here. It is amazing that such a simple rule gives the structure with infinite complexity and detail.