Understanding Julia Sets: Simplified Explanation & Assistance | Expert Help

[dmehling]

I need someone to help explain to me in simple terms how Julia sets work. I understand how the equations governing the Mandelbrot set work, but am finding Julia sets to be a little more complex and difficult to understand.

 

[HallsofIvy]

Actually, finding Julia sets requires exactly the same computation as the Mandlebrot set.

Both involve the iteration $z_{n+1}= z_n^2+ c$. With the Julia sets you are given a fixed "c" (and we refer to the Julia sets as "J_c") and the Julia set consists of all $z_0$ such that that iteration converges. The Mandlebrot set consists of all c such that the iteration with $z_0= 0$ converges. (Some texts say "the iteration with $z_0= c$". Of course, if you start with $z= 0$ you immediately get $z_1= c$ so the convergence is the same either way.)

If I wanted to draw the Julia set, J_c, I would set up a double loop to step through every possible z₀= x+iy and check each to see if the sequence converges. If I wanted to draw the Mandlebrot set, I would set up a double loop to step through every possible c= x+ iy and check to see if the sequence starting with z₀= 0 converges.

The Mandlebrot set, by the way, "indexes" the Julia sets. If c is a complex number well within the Mandlebrot set, then J_c will be a single "blob" with boundary less complicated the deeper inside it is (If c= 0, J₀ is simply a disk). If c is near the boundary of the Mandlebrot set, J_c will be a single connected set with a fractal boundary. If c is just outside the boundary of the Mandlebrot set, J_c is a number of disjoint piecess. If c is far outside the boundary of the Mandlebrot set, J_c is a "dust".

Also, while Mandlebrot worked for IBM, Gaston Julia lived around the beginning of the 19 th century and did all computations by hand!