TITLE: What exactly are fractals
QUESTION [48 upvotes]: I have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of this, but sadly Wikipedia hasn't been of much help. I have a few questions to get me started on this then.

What does $z_{n+1} = z_n^2 + c$ mean? Do we take an initial value of $z_n$ and calculate successive points and go on plotting them on the complex plane?   
What is its historical significance? I feel that knowing where something came from helps us appreciate it even more. Where did this equation first come from? Why was it required to be studied? 

Those are my specific questions for the moment, but considering the fact that I am only trying to learn about fractals, I may not be very well equipped to be asking the right questions, in which case you could tell me anything else you think is worth mentioning. 

EDIT: I have one specific concern about the equation. It says that $z$ and $c$ are complex numbers. All very good, apart from the fact that $\mathbb R$ is a subset of $\mathbb C$ so apparently the starting values can be real too. But wouldn't that lead to all points falling on the real line, and leading to a plain old line instead of the Mandelbrot Set that we know? Although I couldn't find any references(hence this question) wouldn't simply defining the initial values to be non-real be a little... arbitrary? I think I might have a wrong idea of what equation actually means then.

REPLY [4 votes]: Where does $z_+=z^2+c$ come from?
Actually, this story began with the study of Newtons method. If you apply it to a polynomial equation, you get an iteration of a rational function.
From that the topic was generalized to arbitrary rational functions with no connection to Newton's method and then specialized to the "easy" cases.
Linear
The first being linear functions. $z_+=ax+b$. From the solution theory of linear recurrences on knows that one can multiply by $(a-1)$ and distribute $b$ to obtain $$w_+=(a-1)z_++b=a((a-1)z+b)=aw$$ to get a normal form and to know that for $|a|<1$ it is a contraction to zero and for $|a|>1$ it diverges to infinity.
Quadratic
Next simple are quadratic iterations $z_+=az^2+bz+c$. Again, shifting and scaling of the sequence will not change the quality of the picture, so one may first try to reduce the number of free coefficients. Multiplying by $a$ gives $$az_+=(az)^2+b(az)+ac,$$ so wlog. one may set $a=1$. Now complete the square,
$$
az_++\tfrac b2=\left(az+\tfrac b2\right)^2+\tfrac b2-\tfrac{b^2}4+ac
$$
So set $w=az+\tfrac b2$ and $\tilde c=\tfrac b2-\tfrac{b^2}4+ac$, and the reduced form of the quadratic iteration is
$$
w_+=w^2+\tilde c,
$$
the iteration of the Mandelbrot fractals.
There is another normal form of this iteration, the Feigenbaum iteration, that is usually only considered on the real line, more specifically on the interval $[0,2]$, with a parameter $λ\in[0,4]$,
\begin{align}
x_+=λ x(1-x)
&\iff 
(-λx_+)=(-λx)^2+λ(-λx)
\\
&\iff 
(-λx_++\tfrac λ2)=(-λx+\tfrac λ2)^2+\tfrac14-\tfrac{(λ-1)^2}4
\end{align}
so it covers the real line in the Mandelbrot diagram from $-\infty$ to $\tfrac14$, with the $λ\in[0,4]$ covering the range starting from $-2$, the point of the antenna of the Mandelbrot set.
Cubic
One can now continue with cubic iterations. There one can again normalize the leading coefficient to $1$ and the quadratic to $0$, so the reduced form is
$$
z_+=z^3+sz+t
$$
but now we have 4 real degrees of freedom and a symmetry, in that changing $z$ to $-z$ results in the equally reduced iteration
$$
z_+=z^3+sz-t
$$