Fermat's principle of least time...

It is amazing how things in nature works the way it does. I recently came across this rather interesting concept and thought that it was pretty cool. Fermat is one of those physicists who lived in the early 17th century. He discovered that light always takes the path that corresponds to the shortest time. What does that mean?

Assume a circle which is divided along its diameter such that there is an upper and lower half. Choose two points on the circumference of the circle in the upper half (say points A and B). Assume that there is a mirror along the axis of the diameter such that any light from a source A gets reflected at the diameter and reaches the point B. There are several ways the light could get reflected. But, in reality, the light gets reflected at a point such that the distance from point A to mirror to point B is the shortest. It so happens that, at that point, the angle of incidence is equal to the angle of reflection. It doesnt stop here...What about refraction?

In the case of refraction, assume all the conditions as above, except that instead of the light getting reflected by the diameter axis, it is going to pass through it and hit the other side of the circle at a slightly different angle. If the lower half of the circle is a denser medium (say glass/water), the ray of light is slightly shifted inward and reaches a point B'. If you connect the line AB', that would be the shortest distance. But then, light doesnt follow that path. Why? Since light travels slower in the denser medium, it tries to trace a lengthier path in the air and a shorter path in the denser medium. In effect, light reaches the point B' in the shortest time. The factor that accounts for the density is nothing but the refractive index. Quite amazing, isnt it? This formed the fundamental concept through which major discoveries like Calculus and Wave theory were made later. If I think about it, it makes me wonder if everything in nature has an intent. Hmm..that would be a topic for another discussion.