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

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