The set of points in a rational polygon P not illuminated by any x in P is finite

You need to know: Polygon, angles measured in radians, angle of incidence, angle of reflection.

Background: Let P be a rational polygon, that is, polygon whose angles measured in radians are rational multiples of . For any point (called a light source), consider a light ray starting from x and moving inside P, with the usual rule that the angle of incidence equals the angle of reflection. A point is called illuminated if there is a light ray starting from x which passes through y.

The Theorem: On 10th July 2014, Samuel Lelievre, Thierry Monteil, and Barak Weiss
submitted to arxiv a paper in which they proved that in any rational polygon P, and for any point there are at most finitely many points which are not illuminated from light source x.

Short context: In 1969, Klee asked whether any point x in any polygon P illuminates the whole P. In 1995, Tokarsky answered this question negatively by constructing an example in which one point y is not illuminated. The Theorem states that, at least for  rational polygons, only finitely many points can remain unilluminated. It is just one out of many corollaries of deep “Magic Wand Theorem” proved in 2013 by Eskin and Mirzakhani.

Links: Free arxiv version of the original paper is here, journal version is here.

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