I've been looking at the Mandelbrot set and playing around with the powers on it. In the sense that I know the equation is $z^2 +c$, and I've been changing up the $2$. As it happens gradual increases in the power have interesting effects, but I can't find much about them online. Why is it that the set uses the power $2$ as opposed to anything else?
Also, on a side note, does anyone know an explanation as to why the Mandelbrot set is shaped the way it is? With the different sections in it, and the symmetry about the real axis?
Thanks :)
Why? Well, because the Mandelbrot set is defined to be the set of parameters $c$ for which $z^2 + c$ does not escape to infinity. Quadratic polynomials are the first simplest example of functions whose iterations give an interesting dynamics, hence they are widely studied.
If you change power of the polynomial, you are going to get a different dynamics and and a different set in the parameter space.
What follows is an overview of facts about the Mandelbrot set, as an exhaustive explanation would be worth a long book.
About symmetries and properties: A lot of properties of the Mandelbrot set are currently being studied and they stay open questions (e.g. MLC, the local connectivity of the set), while many others are known. Lot of interest in dynamics is about the measure and/or dimension of the set and its boundary, for instance.
The Mandelbrot set is symmetric with respect the $x$-axis, simply because you can prove that if $c \in M$, the Mandelbrot set, then $\overline{c} \in M$ as well.
Also, the set is often divided in different components (for instance, hyperbolic components) depending on the nature of the fixed point. The main cardioid is made of polynomials $z^2+c$ for which the fixed point is attracting. Hyperbolic components are conjectured to be dense, but that is also an open problem (related to the above mentioned MLC conjecture).
Regarding self-similarity, the Mandelbrot set is not self-similar. Several progresses have been done until recent years, to determine around what points it is actually self-similar. Renormalization gives you that the Mandelbrot set is actually quasi-self-similar, i.e. it contains copies of itself (often called "baby Mandelbrot sets") at any scale. Note that this result is highly non-trivial and extremely fascinating.
A quick googling led me to the Multibrot set, which you might be interested in.