Julia Set

Suppose p(z)=z²+c is a complex quadratic polynomial (the value c is a given complex number).  A Julia set is created by iterating p(z).  That is, the output of p(z) is fed back in as input for a new value of z.  Therefore, one can rewrite the polynomial as z_n+1=z_n²+c.  For any initial value, z₀, iterations of p(z) will either be bounded or unbounded. Suppose the set of all z-values that result in bounded iterations of p(z) is denoted by B.  Let the set of z-values that result in unbounded iterations of p(z) be denoted D.  It turns out that the sets B and D intersect.  The intersection of these two sets is called a Julia set.

See the Matlab code used to generate the picture above.  Varying the value of c in p(z) results in different Julia sets.