Appears in collection : Mathematics on the Crossroad of Centuries - A Conference in Honor of Maxim Kontsevich's 60th Birthday

What are the slopes of periodic billiard paths in a regular polygon? We will connect this question and others to: - cusps of thin groups, - curves on Hilbert modular varieties, - heights from Jacobians with real multiplication, and - a spectral gap for the Galois orbits of triangles. Along the way we will encounter issues of chaos and decidability, first appearing in polygons with 7 and 12 sides.