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Oct 12, 2014 at 23:26 comment added Tony Huynh You're welcome, Joseph. Note that it is not inconsistent with what I said that someone has proved that rational pentagons are dense (but I am unaware of such a result). If I had to bet a dollar, I would wager that it is false for pentagons. I think the consensus is also that the set $S$ does not exist, but these kinds of problems are notoriously difficult.
Oct 12, 2014 at 23:14 comment added Joseph O'Rourke Thank you, Tony. Remarkable that even the density question for pentagons remains open (although apparently solved by Mordell for quadrilaterals).
Oct 12, 2014 at 22:59 history edited Tony Huynh CC BY-SA 3.0
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Oct 12, 2014 at 22:54 history answered Tony Huynh CC BY-SA 3.0