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flawr
flawr
9:43 AM
would be even crazier if $\Lambda >0$ :)
 
 
5 hours later…
flawr
flawr
2:44 PM
@NathanMerrill you might like this: i.redd.it/xr2pw7k7tbb01.jpg source
 
Nathan Merrill
Nathan Merrill
3:21 PM
that is good :)
 
 
2 hours later…
El'endia Starman
El'endia Starman
5:22 PM
@flawr I saw this yesterday and loved it.
 
 
5 hours later…
flawr
flawr
10:25 PM
Just read this question on MO
1
Q: A Voronoi Iteration Game

Hauke ReddmannLet $P_i$ 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=...

gt.geometric-topology
can anyone find an example where |P_{i+1}| >= |P_i| for a finite set?
 
PhiNotPi
PhiNotPi
10:40 PM
"Map" of the solar system
 
Martin Ender
Martin Ender
11:29 PM
@flawr (0, 0), (4, 0), (0, 4), (4, 4), (2, 2), (3, 2), (1, 2) gives 8 vertices in the Voronoi graph
I'll post it as a comment as well.
Iterating on those Voronoi vertices even gives you a diagram with 9 vertices.
 
flawr
flawr
I did not expect that:)
I tried to prove that |P_{i+1}| < |P_i|, but only managed to prove |P_{i+1}| <= 3|P_i|
 
flawr
flawr
11:52 PM
@MartinEnder Did you have some strategy for constructing this example?
another thought: Given the vertices (or even the full graph) of a voronoi diagram, is it possible to reconstruct the original points?
@flawr No in general it is not, consider the case [(0,0),(0,1)]
 

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