what is Mandelbrot set ?

Question

what is Mandelbrot set ?

In: Mathematics

Answer 1

To really understand the Mandelbrot set you’ve got to know a bit about complex arithmetic and complex algebra. The Mandelbrot set is a map of a certain property of how multiplication and addition affect each other in the complex numbers.

Specifically, it asks “If you start with a number and square it, then add the number, then square it, then add the number, then square it, then add it, etc etc etc, what happens?”

First you can consider this question using only the familiar set of “real” numbers. If we suppose our starting number is 0.2, then following this square-and-add process for a few steps, we get:

0.2^2 = 0.04
0.04 + 0.2 = 0.24
0.24^2 = 0.0572
0.0572 + 0.2 = 0.2572

and onward like that. Now, depending on your starting number, a couple of different things can happen. if a number is between 0 and 1, then squaring it gives you a smaller number than you started with. If you chose a very small number, then the squaring step will make the number even smaller, and this shrinking in the squaring step will keep the numbers quite small, *despite* the fact that the adding step makes it a bit bigger each time.

Alternatively, you could choose a bigger starting number like 2. 2 squared is 4, then you add 2 which makes 6, and then 6 squared is 36, and… things blow up fast.

Now, things act a little more strangely when you try using a negative number. -1 squared is 1, then you add -1 and get 0, then you square that and get 0, and then you add -1 and get -1, and you square that and get 1 again, and you’re in a loop!

So those are our options: things converge toward a small number, they blow up into increasingly large numbers, or they end up in a stable cycle.

The Mandelbrot set is what happens when you allow your starting number to be positive, negative, *or complex*. Complex numbers are like 2-dimensional numbers. Where multiplying real numbers can be visualized as stretching or squashing the number line, multiplying complex numbers is like stretching or squashing *and rotating* the coordinate system. This can be pretty hard to visualize but there are a lot of great [videos](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-mul-div-polar/a/visualizing-complex-multiplication) with animated diagrams that can help. It gets pretty, eheheh, complex.