Please consider the following recursive diagram: diagram
Each triangle is connected at the midpoint of a side to the corner of an inner triangle which is 1/4 times the size. The total line length of the figure is convergent.
The puzzle question that goes with the figure is: starting at the top, what is the shortest distance we have to walk to cover the entire figure?
What I would like to know is: is the answer to this question is well defined? We can use a recursive procedure, but to go the shortest distance, we must:
- Go along the edges until we can cross to the inner triangle.
- Do the same procedure for the inner triangle.
- Cross to the outer triangle again and finish going around the edges.
Is this simply a kind of a Zeno's paradox? To me it seems that when the procedure is changed to first finish the outer triangle completely before moving inwards, there is no problem: we can keep going and the total distance is convergent. But by the first method, we first have to do an infinite number of things, and then finish up the outer triangle.