Mandelbrot and Julia Sets
Date: 06/29/98 at 18:10:34 From: Paulo Jorge Matos Subject: Fractal equations (Mandelbrot and Julia) Hi Dr. Math, I started studying fractals on my own because I will only learn complex numbers in Math next year. I'm studying to be a software engineer and I know that Mathematics is a very important subject to be a good software programmer but I also know that it is not easy. If anybody asks you about a good book to start understanding fractals, I'd recommend the following. I've read it. It's my first about fractals: _The Magic Machine: A Handbook for Computer Sorcery_ by A. K. Dewdney. I've made a program to build the Mandelbrot set, but I'm having some difficulty with the Julia set. I know that both equations are z = z^2+c, where z and c are complex numbers. Computers just don't get complex numbers, so I after some equations I got to the following to the Mandelbrot set if z is composed by x + yi and c is composed by a + bi: x = x^2 - y^2+a y = 2xy + b But for the Julia set I have reached wrong equations. I've reached the following: y = b - 2ab x = a - (a^2 - b^2) But it is not possible because I have to iterate x and y, and they need to appear also on the right part of the equation. What's wrong? Is it too confusing? Hope you can help. Regards, Paulo Jorge Matos
Date: 07/01/98 at 22:56:11 From: Doctor Pete Subject: Re: Fractal equations (Mandelbrot and Julia) Hi, The difference between the Mandelbrot set (M-set) and Julia set (J-set) is not in the formula, but in the type of iteration involved. The complex number c is called a parameter. The M-set is plotted in the *parameter space* of c, whereas the J-set is not. It is instead plotted in the orbit space z. That is, for the M-set, you calculate: 0, c, c^2 + c, (c^2 + c)^2 + c, ... for a given complex number c, and see if the sequence converges. If it does, that point c in the complex plane is in the M-set. So what you're plotting is a picture of all complex c such that the mapping z |-> z^2 + c is bounded, where the initial condition is z = 0. However, the J-set is plotted for a *single* value of c, but you are now changing the initial condition: z, z^2 + c, (z^2 + c)^2 + c, .... The value of c is fixed for all points z. So in a J-set you're plotting in the space of starting values of z. In short, you plot the M-set by iterating over many different values of c, whereas in the J-set you fix c and iterate over many different values of z. In both cases, the formula for iteration is the same; only two things change: M-set: starting condition is z = 0 pick a point in the plane, c iterate and see if bounded; if so then c is in the M-set J-set: fix a number c -- this never changes pick a point in the plane, z iterate and see if bounded, if so then z is in the J-set. So there are many J-sets, one for each c, but only one M-set. If you think about it, the M-set is a "guide" to all the J-sets. What I mean is that if a point c is in the M-set, then the J-set with starting value c is connected. This is because there is a theorem which says the point z = 0 is in a J-set if and only if that J-set is connected. For an online introduction to fractals, please see Paul Bourke's page: http://astronomy.swin.edu.au/~pbourke/fractals/fracintro/ - Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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