An Unexpected Solution

Oftentimes, math is distilled down to the task of answering questions. But in doing so, it sometimes loses what I like to think it truly is: the art of asking the right questions (and this isn't just me, this is actually a famous quote by Cantor, a renowned mathematician). And it was during a recent math class that this idea truly came to life.

It starts with a perfectly innocent question: suppose a person left their phone on a beach, and they are currently swimming in the water. They'd like to get to the phone as quickly as possible, and they are a little slower in the water than on land. What path should they take to get there the fastest? The idea here is that taking a 'straighter' path in the water would save them some time.

The textbook solution was just 'do a bunch of calculus to get the answer.' But there's actually a beautiful hidden link to Physics just waiting to be explored (spoiler alert: Bernoulli's solution to the Brachistochrone problem).

It turns out that there's something similar in physics, when light bends when it changes media: the concept of refraction. A common axiom of optics is that 'light takes the path of least time.' But that sounds familiar, doesn't it?

By treating the speeds as refractive indices of light, then using Snell's law (something almost all of us learn in 8th grade), we can arrive at a beautiful, 2-line solution, that also brings to light (no pun intended) another hidden connection with trigonometry. How fascinating!

This is precisely what I love about math - the fact that it hides within a treasure trove of innumerous such unexpected relatioships to other areas of math and the natural sciences. In many ways, math is the study of logic and connections more than it is that of numbers.