Refraction of light marching band analogy When trying to understand the refraction of light when it hits a slower medium, lots of people seem to be enlightened by the 'marching band' or 'marching soldiers' analogy, which 'explains' that when a platoon marching in a straight line on a concrete road hits the side of the road at an angle and goes into a swamp, the soldier which is first to hit the swamp will start walking slower while the soldiers that are still on the road keep going at the previous speed. This supposedly explains intuitively why the marching band changes direction.
I don't understand how this works at all, the soldiers hitting the slower medium will start walking slower but why on earth would they change direction as well? As I see it, the front line of soldiers will form a different angle to the direction the platoon is marching in after it has hit the interface, but the platoon should still be going in the exact same direction even if the soldiers run into each other because the guy in front is slowing down. Am I missing something in the analogy or is the analogy a really bad one (which I'm starting to suspect strongly)?
 A: Imagine not the direction of the column but the direction of the front row. Suppose the front row of soldiers were carrying a horizontal bar, the one on the left hitting the swamp would slow down while the one on the right was still moving quickly so the bar (=wavefront) would change direction
It's a slightly bad analogy. A much better one is: 
Imagine you were on a beach and had to reach somebody in the water. You could run to the nearest point of the water and swim diagonally, but assuming you swim slower than you run, this would take the most time. Or you could run all the way along the beach to the closest point to the person and swim the shortest distance but this is still a longer than necessary path.
But there is a point on the beach where if you ran to that point then swam, you would minimise the total time taken. The point would depend only on the ratio of your speed running and swimming. This is exactly what light does. It's called Fermat's principle and is one of the most fundamental things in physics.
