Fractal Theory: Connectedness
It's quite obvious that the outer bands around the Mandelbrot set form complete loops around the Mandelbrot set. The band representing two iterations, for example, travels smoothly around the outer edges and then connects back up to itself. There are no other points that have an iteration count of two except on this band, and all the points on this band are connected by other points with an iteration count of two. This is less obvious but equally true for all other bands. If you zoom in to the band that represents ten iterations, you can "drive" all the way around the Mandelbrot set, following that band, and return to where you started. You could try it for the band representing 100 or 1,000 iterations, but it would take a very long time.
This little idea, of all the bands being single bands going all the way around, doesn't seem too amazing when you look at the outside of the Mandelbrot set. But when you've zoomed in 20 times and are looking at a complex shape surrounded by spirals, it's quite amazing to think that each band you see somehow works it way around each of the individual nodes in the structure, into and out of all the arms in the spiral and then onwards around the Mandelbrot set, without ever crossing another band or ever quite disappearing.
The Mandelbrot set, the internal black (by default) area, itself is not excluded from this rule. Whenever you come across a miniature copy of the Mandelbrot set (and there are an infinite number of them scattered around), you can be sure that this tiny copy is connected to the main Mandelbrot set by one, and only one, infinitely thin filament that you will never see, but whose presence can be detected by seeing the constantly thinning bands of colour squeezing in on it from both sides.