"The first notion here is that of a Julia set of quadratic iteration. Pick a point C with coordinates u and v, and call it a "parameter." Next pick, in a different plan, a point P₀ with coordinates x₀ and y₀. Then form x₁ = x ₀² - y₀² + u and y₁ = 2x₀y₀ + v. These formulas may seem a bit artificial, but they simplify if the point C with coordinates x and y is represented by a complex number z = x + iy. (One can add and multiply complex numbers like ordinary numbers, except that i² must always be replaced by -1,) In terms of the complex numbers C = u + i v and z = x + iy, the preceding rule simplifies to z = z₀² + C and (more generally) z_k+1 = z _k ² + C . But even the reader who is scared of complex numbers will understand the expressions in terms of x _k and y _k.

"When the orbit P _k fails to escape to infinity, the initial P ₀ is said to belong to the "filled-in Julia set." An example is

If you start outside of the black shape, you go to infinity. If you start inside, you fail to iterate to infinity.