The Sierpinski triangle can be "generated" by the algorihm where you start in the triangle, pick a vertex at random, then move half the distant towards it, draw a dot and then repeat this.

I wasn't able to find anything on whether doing this for other polygons produces similar sort of structures, so I tried it out and this is what I got:

![enter image description here](https://i.stack.imgur.com/pIsey.jpg)

After these there just seems to be a small hole in the middle. Well the square isn't interesting. But what about the holes in the pentagon and the looming star structure in the hexagon? Are these of any interest?