Fractals and complex numbers:
Big ideas
Each complex number (c) makes a new rule for how to make a Julia set. Finding the Julia set involves iterating a complex function (doing it over and over), and looking at whether the outputs stay small, or get very large. Numbers (z) that stay small when you iterate them are in the Julia set; numbers that get large when you iterate them are not in the Julia set.
If you look at each Julia set, and the number that makes its rule (c), then if the Julia set is connected, then that number (c) is in the Mandelbrot set. If the Julia set is in several pieces, then that number is not in the Mandelbrot set
If you iterate 0, and it stays small, then the whole Julia set is connected. If you iterate 0 and it stays large, then the set is disconnected.
Sample questions:
1. Find the next Julia function iterations, using the function f(z) = z2 + .5 - .3i starting with the point z0=0
2. Find the next Julia function iterations, using the function f(z) = z2 + .2 - .6i starting with the point z0=0
3. Find the next Julia function iterations, using the function f(z) = z2 + -.7 + .5i starting with the point z0=0
4. If these are some Julia iterations, what can you say about the Julia set?
A. z0 = 0, z1 = -.2 +.3i, z2 = -.25 + .18i, z3 = -.17 + .21i, z4= -.21 + .23i , z5 = -.21 + .20i
B. z0 = 0, z1 = .5 +.1i, z2 = .745 + .2i, z3 = 1.01 + .40i, z4= 1.36 + .903i , z5 = 1.54 + 2.54i
A. z0 = 0, z1 = .2 +.6i, z2 = -.12 + .848i, z3 = -.49 + .401i, z4= .28 + .21i , z5 = .24 + .72i