Snell’s Law → BrachistochroneLight always takes the quickest route. Drop a ray through a stack of thin layers, each one letting it travel faster than the last, and Snell’s law kinks it a little at every boundary. Add more layers and the zig-zag smooths into one graceful arc — a cycloid, the exact curve a bead slides down in the least possible time.Johann Bernoulli’s 1696 trick: replace the falling bead with a ray of light in a medium whose speed grows as √depth. Fermat’s principle then fixes sin θ / v = constant at every layer; send the layer count to infinity and that refraction law integrates exactly into the brachistochrone cycloid.