# Are the vertices of a Voronoi diagram obtained from a Sierpinski attractor also a kind of attractor?

Trying to understand how the Voronoi Diagrams work I did a test generating the Voronoi diagram of the points obtained from The Chaos Game algorithm when it is applied to a $$3$$-gon. The result is a set of points part of the Sierpinski attractor, the Voronoi cells and the vertices of each cell.

This is a little animation of the first 200 steps of The Chaos Game for a $$3$$-gon and how the Voronoi diagram is generated:

And this is how it looks like a Voronoi Diagram applied to TCG after $$2000$$ steps:

Update 2016/01/25: as suggested by @Mark McClure in the comments, here is the result of the Voronoi diagram applied directly to the Sierpinski gasket (recursion level $$6$$).

I would like to ask the following question:

1. If the Voronoi diagram is using the Sierpinski attractor points as the seeds or generators of each cell, are the vertices then a kind of attractor as well?

2. I was not able to find papers or references about this kind of test (obtaining the Voronoi diagram of an attractor). Are there any papers or references?

3. For what other attractors would be interesting to obtain a Voronoi diagram? (I am working currently in a TCG $$4$$-gon algorithm, I will try to add the results in some days)

I am asking this because the vertices are partitioning the space according to the Sierpinski attractor, so I wonder if they will inherit some properties of the attractor (like being an attractor itself, or some special orientation of the Voronoi poles, etc.).

Thank you!

• I recommend that you try it with a regular image generated by a deterministic algorithm. I think it will probably be easier to see the convergence of your Voronoi vertices in that case. – Mark McClure Jan 15 '16 at 11:15
• @MarkMcClure added, thanks for the suggestion! – iadvd Jan 25 '16 at 4:46
• That does look pretty cool! – Mark McClure Jan 26 '16 at 13:39