Timeline for Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?

9 events

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Oct 30, 2013 at 16:25	comment	added	Jeremy Rickard		@j.c.: I don't mind at all! Very nice pictures! It might be fun to look at some examples where there are two maximal eigenvalues involved, with equal coefficients.
Oct 30, 2013 at 16:01	vote	accept	mathlove
Oct 30, 2013 at 16:01	comment	added	mathlove		@JeremyRickard: Your answer did surprise me. Thank you so much for writing your answer with details. I had a great time to read your answer which was beyond my imagination.
Oct 30, 2013 at 15:09	comment	added	j.c.		Nice answer! I hope you don't mind that I've added some illustrations to your answer.
Oct 30, 2013 at 15:09	history	edited	j.c.	CC BY-SA 3.0	add images
Oct 30, 2013 at 10:23	history	edited	Jeremy Rickard	CC BY-SA 3.0	added 1646 characters in body
Oct 30, 2013 at 9:57	history	edited	Jeremy Rickard	CC BY-SA 3.0	corrected some minor sign error
Oct 29, 2013 at 18:11	comment	added	Jeremy Rickard		It's just occurred to me that the operation doesn't preserve convexity or even the property of being a simple polygon: consider what happens if you start with the convex quadrilateral with vertices at $(-1,0)$, $(1,0)$, $(x,x)$ and $(-x,x)$ for very small positive $x$. I think my answer still works if you interpret the question appropriately, but I'm no longer confident about what happens if $n$ is odd but not prime.
Oct 29, 2013 at 17:44	history	answered	Jeremy Rickard	CC BY-SA 3.0