The light prefers the shortest. Physics and geometry about shortest path problems from Heron to Fermat

International audience<p>Our work is a detailed report about an educational experience, tailored on groups of students of the first two years of high school. The starting point is the classical “Seven Bridges of Königsberg” problem but the whole historical background of our inquiry spans from Heron to Fermat, on the thread of the shortest path problem. In particular, starting from physical phenomena easily experienced in the daily life, the Fermat's least time principle is introduced. The light behavior is described in a geometrical optic approximation, in analogy with kinematics. The activities, based on laboratory teaching and learning, are focused on linking physics and mathematics: refraction and reflection experiments compass-and-straightedge<strong> </strong>constructions and interactive open source geometry software, as Geogebra; recognition of isometries in problems and reasoning based on their features; working by artifacts to build knowledge and skills; translation of geometrical objects in algebraic language.