# How does one determine the containing boundary of a fractal?

In the Mandelbrot set, the fractal is said to be contained in the circle of radius 2. $$z_{n+1} = {z_{n}}^{2} + c$$ I did read about a proof that showed values of 'c' beyond this circle are not bounded and hence the set is contained within.

But say if someone discovers a new set that generates a fractal, How does one determine the containing boundary of that fractal ?

PS: I have studied basic engineering Mathematics and learning fractals on my own

You need to calculate a lot of points and find which points stay finite and distinguish points that take a long time to go to infinity from points that take a short time. Basically use a computer, then look at the result. Also its a fractal, so the boundary can't be determined, it would have infinite points in it. You could approximate it however, use different colors for different escape times.

• Thanks, I understand that. But I'm referring to the bailout/escape value. – Bharath Mar 23 '15 at 7:40
• Every c with an absolute value above 2 is good start. – Zach466920 Mar 23 '15 at 14:12

(I gave the answer above as well) Given the lack of answers to your question I'll link you to this. It should give you more than you can handle, which is why I refrained, I barely understand it myself, but hey maybe I'm projecting. It'll answer all of your questions.

One can you geometric algorithms like boundary checking ( Sobel filter)