Timeline for Which polygons have *simple* periodic billiard paths?
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Feb 8 '15 at 13:20  history  edited  Alexandre Eremenko  CC BYSA 3.0 
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Feb 8 '15 at 4:25  comment  added  Aaron Meyerowitz  The conditions are sufficient for a cyclically ordered list of angles to be realized by some polygons having a polygonal path touching each side. A regular pentagon has such a path but an equiangular pentagon with sides of lengths $1,\epsilon,1.6,\epsilon,1$ seems unlikely to. For a triangle the angles do determine the sides (up to similarity) so acute triangle have paths and obtuse ones do not.  
Feb 7 '15 at 20:50  comment  added  Alexandre Eremenko  Yes, I think these considerations permit to describe explicitly all such polygonal tables.  
Feb 7 '15 at 19:59  comment  added  Will Sawin  It is not too hard to see that the interior angles of the polygon you get by doing this are the averages of the adjacent interior angles of the billiard path. From this you get an explicit system of inequalities that the angles satisfy if and only if there is a polygonal billiard path touching every edge.  
Feb 7 '15 at 15:11  history  edited  Alexandre Eremenko  CC BYSA 3.0 
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Feb 7 '15 at 14:28  history  answered  Alexandre Eremenko  CC BYSA 3.0 