Indexing the archive…
Your Universe of Digital Possibilities
Between two points, light could take any path — but it takes the one that costs the least time. Slowed by the denser medium, it spends less of its journey down there, bending at the surface by exactly the amount that trims the clock. Minimise that time over the crossing point and Snell’s law drops out, unbidden. Nature, here as everywhere, is lazy.
Of all the paths light could take between two points, it takes the one whose travel time is stationary — almost always the least. The speed in a medium is c/n, so the ray bends to spend less length where it would crawl. Every law of ray optics falls out of this one line.
The bend at an interface — not a separate law but the price of least time: minimise T over the crossing point and the derivative vanishes exactly when n sin θ is conserved across the boundary. Past the critical angle there is no real θ2 and the light reflects entirely.
Geometric length weighted by the index — the distance light “feels”. Fermat’s path is the one of stationary OPL; a perfect lens works by making every route to the focus take equal optical path, so they arrive in phase.
Fermat’s least time is the optical face of least action (The Action, INST·63): both pick the stationary path of an integral, and Hamilton’s optical–mechanical analogy made them one. Johann Bernoulli solved the brachistochrone (The Bead, INST·62) by a beautiful trick — he sliced the fall into layers of increasing speed and turned it into thisrefraction problem, letting Snell’s law trace the curve of fastest descent. And where The Well bends light by curving spacetime, The Ray is the flat-space floor beneath it: gravitational lensing is Fermat’s principle run on a metric instead of a medium. One lazy rule — take the quickest path — underwrites them all.