Every triangle with all angles at most 100 degrees has a periodic billiard path

You need to know: Polygon, angles measured in radians, angle of incidence, angle of reflection, limits, Hausdorff dimension of a set.

Background: A billiard path in a triangle T is a point moving inside T with constant speed, with the usual rule that the angle of incidence equals the angle of reflection. Such a path is fully described by point’s initial position and initial direction to move. A billiard path is called periodic if, at some time moment, the point trajectory starts to repeat itself.

The Theorem: On 8th September 2007, Richard Evan Schwartz submitted to the Experimental mathematics a paper in which he proved that every triangle with all angles at most one hundred degrees has a periodic billiard path.

Short context: Billiard path is a simple but fundamental example of a dynamical system. More that two-hundred-year-old triangular billiards conjecture predicts that every triangle has a periodic billiard path. The conjecture is relatively easy for rational triangles (those whose angles are all rational multiples of ), and also for right triangles. In 1775, Fagnano proved it for acute triangles. However, there was essentially no progress for general (possibly not rational) obtuse triangles. The Theorem proves the conjecture for all triangles with largest angle at most 100 degrees.

Links: The original paper is available here.

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