You all know the Barnsley Fern and The Sierpinski Triangle.
I tried to find something similar (to the Sierpinski Triangle) in the disk but all I got was this ring:
What would be some other algorithms that produce some fractal-like structure?
EDIT: The way I generated this picture is the following:
I start from some point of the disk (I guess it doesn't matter where you start, I started from the boundary) and then check what angles you can move if you move the lenght of the radius of the disk to still stay inside the disk. At first I chose an angle from these possible ones at random and that way I seemed to fill the whole disk. Then I narrowed the range for the possible angles (for example $[\pi - \alpha, \pi + \alpha] \to [\pi - \frac{\alpha}{2}, \pi + \frac{\alpha}{2} ]$, where $\alpha > 0$) down and the formation seemed to move towards an annulus. When you narrow the choise of the angle to a single angle ($x + q(y-x)$, when the interval is $[x,y]$, $q \in (0,1)$), it collapses to this ring. I guess this is because this ring is fixed by the algorithm so when you land on it, you stay on it.
I see now my question was perhaps little bit too broad. I think the algorithms I'm most interested in for the moment is these ones where you start from a point and then move along according to some rules like in the Sierpinski triangle generating algorithm. Well there are many ways to generate it, but I think you know what I mean, the one where you move towards the corners.