# Fractal followup followup

Again, after yesterday’s post (which was a response to the post of two days ago), I found a few more cool things with this method of generating fractals.  In particular, we take our point x and a random vertex v, then move our point to ax + bv.  To get the Sierpinski carpet, we have a = 2, b = 2.  To get the sets from yesterday, I used a = 1/(n-1), b = (n-2)/(n-1), where n was the number of vertices.  Today, I share some pictures using a = k/(n-1), b = (n-k-1)/(n-1).  I am not sure whether each of these is a fractal (though I suspect they are), but there is something going on (note- Leonid Kovalev pointed out yesterday that Jeremy Tyson and Jang-Mei Wu have a paper (pdf link) investigating sets generated like this):

5 vertices, with k = 2. Used 500,000 points for this one, I think most of the pentagon is meant to be in the set...

6 vertices, with k = 2.

7 vertices, with k = 2. Looks a little like the chainring on my bike...

16 vertices, k = 1

16 vertices, k = 2

16 vertices, k = 3

16 vertices, k = 4