Understanding the mandelbrot set equation(s)

[ChrisBaker8]

I'm not sure if this is the correct section to be posting in. I'm writing a summary of the mandelbrot set and I'm not sure I understand how the points are calculated.

I've got the equation:

z = z^{2} + c

This means each value is squared, and then a constant value c is added, to get a new value of z, which then goes through the same process. What does this infinite series represent? Is c an actual defined constant, like pi or G, or is it just a value that can be chosen?

 

[seycyrus]

C is a chosen parameter.

The set is usually colored related to how "quickly" the iteration goes to infinity.

 

[Count Iblis]

If you fix c then you don't get the Madelbrot fractal. What you get is the Julia fractal. This then depends on the value of c you've chosen. Now for some values of c the Julia fractal will be totally disconnected. In tat case, you'll the fractal looks like consisting of islands but if you zoom into these islands, you see that each island consits of smaller islands which in turn consit of even smaller islands and ultimately everything consists of isolated points.

For some other values of c, the Julia fractal is a connected structure. Now, you can prove that a Julia fractal is connected if and if only if the point z=0 belongs to the Julia set. What Madelbrot did was to look at the set of all the values of c for which the Julia fractals are connected.

So, given some C you look if the series:

z_1 = 0^2 + c = c

z_2 = z_1^2 + c

z_3 = z_2^2 + c

etc.

goes to infinity or not. The set of all these values for c is the Mandelbrot set, which is itself a fractal. The algorithm is thus similar to that for the Julia fractal, except that you take c to be z_1.

 

[HallsofIvy]

By the way, while Benoit Mandelbrot worked for IBM and used the latest computers in his work, Gaston Julia wrote about what are now called "Julia sets" in 1918 and did all computation by hand!