I was asked an interesting question by my tutor today.
We had just finished working out a question concerning a tunnel drilled straight through the earth (ignoring the liquid core, the intense heat and all the other reasons it would be completely impossible to create such a tunnel). If one were to drop a stone though such a tunnel, it would accelerate, pass straight through the core and then start decelerating on its way back out. Assuming the earth to be a perfect sphere, and a complete absence of friction, the stone would forever pass back and forth through the tunnel.
Our tutor then asked us to consider a shorter tunnel, still passing through the earth but not through the center. Would the time it takes the stone to travel from one end to the other be shorter, or longer? At first it may seem obvious that the time would be shorter, due to the reduced length of the tunnel. It may also seem obvious that the time would be longer, since gravity would be weaker, due to more of the mass being one side of the stone than the other.
It turns out that for any two tunnels, the time would always be EXACTLY the same.
I think it's a beautiful example of how ingrained maths is within nature.