A Voronoi Iteration Game Let $P_i$ be a set of points in the plane and $P_{i+1}$ the corners of the Voronoi diagram of $P_i$. Start with some $P_1$ and iterate away. What happens? Can you choose a $P_1$ so the iteration goes into a stable loop? (If you play the game on a torus, $P_1=${some point} works, on a sphere $P_1=${a tetrahedron} works either.) It's relatively easy to see that if $P_i$ has exactly two inner points, $card[P_{i+1}]=card[P_i]$, but this is neither necessary nor sufficient - nothing tells you how many inner points $P_{i+1}$ has.
 A: Not answering your questions, but here are five iterations starting with $10$ random points:

          


          

In the first frame, the $10$ points are clustered in the large central dot.


          

Subsequent frames alternate Voronoi sites (orange) and Voronoi vertices (blue).


And here is the last frame enlarged:

          


          

Enlarged view of 5th frame in sequence above.


          

In the next, 6th frame, the blue points would become the orange Voronoi sites.


It appears to me, from limited empirical exploration, that,
for random points uniformly distributed in a square, the number of points
$|P_{i+1}|$ is very roughly $1.8 |P_{i}|$ with each iteration.
So growing from $10$ points to ${\sim}100$ points after $4$ iterations.
A: In a Euclidean space, the set $P_{i+1}$ is contained in the interior of the convex hull of $P_i$. Thus the convex hull of $P_{i+1}$ is a proper subset of the convex hull of $P_i$, and there can be no cycle.
EDIT: no, sorry, this is false. If $P_i$ are the vertices of an obtuse-angled triangle, then the (only) Voronoi vertex is the circumcenter which lies outside of the triangle. Still, I hope that there is some kind of "monotonicity" that prevents a cycle...
