Rational viewing points in a polygon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:57:01Z http://mathoverflow.net/feeds/question/114079 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114079/rational-viewing-points-in-a-polygon Rational viewing points in a polygon Sinai Robins 2012-11-21T17:20:06Z 2012-11-21T17:30:23Z <p>We refer to the question posed in <a href="http://mathoverflow.net/questions/112714" rel="nofollow">http://mathoverflow.net/questions/112714</a>, but now ask for constructions or for the existence of rational viewing points. We'll call a point $p$ inside (or on) a polygon $P$ a rational viewing point if all of the angles formed by $p$ together with any two adjacent vertices of $P$ are rational multiples of $\pi$.</p> <p>Problem 1. Suppose we have a convex polygon $P$, and there exists a rational viewing point in (or on) $P$. Must there exist infinitely many rational viewing points in (or on) $P$ ? </p> <p>Problem 2. We observe that whenever we have a vanishing sum of roots of unity, with any real coefficients, then we may consider the individual summands as vertices of a polygon $P$, with the origin as a rational viewing point. In this case, are there always other rational viewing points in (or on) $P$ ?</p>