I first envisioned this measuring the distance walking through a city.  Because of the layouts of the streets there is no straight line for the shortest course.  Without trimming any corners in streets, a meandering route through the city would be nearly the same distance traveled as a right triangle from point A to B.  I'm just a Physics major and have had this idea in my head for quite some time and can't seem to get a real good answer to it anywhere on the net so I'd thought I'd try here.  Next, I converted this to paper folds:

Take an isosceles right triangle and "fold" the right angle towards the hypotenuse until it just touches.  This leaves 2 identical triangles that are scaled in size but also leaves an outer "edge" that is equal in length to the two sides.  Repeating this again leaves 2 identical triangles for every original triangle prior to the fold but does not change the outer length from that of the original two sides.  Repeating this to infinity begins to resemble the smaller and smaller pieces of the Fractal Staircase, but in the end one is left with two identical "lines" of seemingly equal length but one is actually root 2 longer than the other. 

Is this an example of fractals and if not, is this a problem that is at least interesting?