Question

Why does the Mandelbrot Set look the way it does? Is it possible for this set to have a different shape, such as an octagon?

Answer 1

The classical Mandelbrot set is for the function f(z)=z^(2)+c. Different functions give different shapes. E.g. for f(z)=z^(3)+c, the Mandelbrot set is symmetric left-to-right as well as top-to-bottom.

Answer 2

What do you mean?

The mandelbrot set is the set of all complex numbers c for which if we start at 0 and iterate the function f(z) = z^2 + c the sequence of numbers we get stays bounded. It is this one thing and it has the one definite shape we all know.

We can define lots of different sets in similar ways that can have all kinds of shapes, but those are not the mandelbrot set.