A recent Youtube video from a channel called Vsauce caught my attention. In the video he mentions this line created by a specific set of geometric events. This line is quite special in its properties, if made into a 3D object it is the fastest path from one point to another. This is due to the perfect balance between distance traveled and velocity.
In this video you can see the three different paths that are built, the linear path, the Brachistochrone path, and the 'extreme' path as they call it. The linear path has the least amount of distance traveled, but also at a slower rate then the others. The 'extreme' path has the greatest speed, but also the most distance to travel. And the Brachistochrone path has the perfect mix of speed and distance to travel making it the fastest.
Another interesting property of the Brachistochrone curve is that no matter where the ball starts on the path it takes the exact same time to reach the bottom.
Here is the link to the full video if you are interested in the geometry part of it too.