If you place a ball at the top of this slope, it will of course roll down. But over what slope would the ball roll down fastest? This simple question has a surprisingly complex answer! You might argue that the straight line on the right-hand side is the shortest route, therefore the fastest. You could also argue that as steep a slope as possible would give much more speed to the ball, enabling it to reach the end faster.

The answer, however, is the slope on the left-hand side, known as a “Brachistochrone curve” (brákhistos khrónos literally means “shortest time”). It is a “cycloid”. To see what exactly that is, you can imagine a wheel. Attach a pen to the edge of the wheel, then roll the wheel along a cloth. The line drawn by the pen on the cloth is a cycloid. Why exactly is this the quickest route for a ball? To find the answer to that question, you will have to pursue a few mathematical disciplines, because geometry, integrals and differential equations are all at play here…

Try to race two balls! How much faster is the brachistochrone curve?