TITLE: Is it possible to plot a graph of any shape?
QUESTION [51 upvotes]: In school, I have learnt to plot simple graphs such as $y=x^2$ followed by $y=x^3$. 
A grade or two later, I learnt to plot other interesting graphs such as $y=1/x$, $y=\ln x$, $y=e^x$.
I have also recently learnt about trigonometric graphs and circle equations.
In the internet, I have seen users posting graphs of different shapes like a heart-shaped graph and a Batman logo-shaped graph.
I am sure there are numerous more graphs that I have yet to see.
Seeing that the graphs can be shaped into shapes like the Batman logo and a heart brings me to my question: Is it possible to plot a graph of any shape regardless of its complexity? Perhaps, shaped into an outline of a person or a landmark? Why or why not?

REPLY [24 votes]: Is it possible to plot a graph of any shape regardless of its complexity?

Definitely not, simply because it is possible to define shapes that are so complex they cannot be computed. More precisely, there exist uncomputable functions $f$, such that no program can compute their graphs $\{(x,y)\mid y=f(x)\}.$ 
In fact, most functions that map $N \to N$ are uncomputable, in the sense that uncountably many are uncomputable, whereas only countably many are computable (there being uncountably many such functions altogether).
An example would be $f(x)$ defined as the number of $x$-state Busy-Beaver-class Turing machines.
NB: This is contary to all $5$ of the other answers -- perhaps because they assume the graph is supposed to be finite (and hence computable).