The Mandelbrot Set and Julia Sets

Complex Iteration

For example, suppose (a, b) = (1/2, -1/2) and (x0, y0) = (0, 0). Then

x1 = x02 - y02 + a = 02 - 02 + 1/2 = 1/2 y1 = 2⋅x0⋅y0 + b = 2⋅0⋅0 + (-1/2) = -1/2

Next,

x2 = x12 - y12 + a = (1/2)2 - (-1/2)2 + 1/2 = 1/2 y2 = 2⋅x1⋅y1 + b = 2⋅(1/2)⋅(-1/2) + (-1/2) = -1

Then,

x3 = x22 - y22 + a = (1/2)2 - (-1)2 + 1/2 = -1/4 y3 = 2⋅x2⋅y2 + b = 2⋅(1/2)⋅(-1) + (-1/2) = -3/2

and so on.

Return to Complex Iteration.