So, I thought of something the other day that my friend once told me when I was in high school. I still remember it, but can't remember what he taught me about it, and can't find anything on Google. I thought I'd post it here to see if anyone had any thoughts.
Basically, imagine you had a square with sides equal to , and you want to travel diagonally from the top left corner of the square, , to the bottom right, .
Now, the shortest route would be to go diagonally, and you can calculate that from Pythagoras' theorem as .
Alternatively, you could go across from , to the top right corner, then down to , and will obviously have travelled .
Now, instead, imagine going halfway between and the top right corner, then down to the centre, then across to the centre of the right hand side vertical line, then down to . You'll again have travelled . If you then repeated this, but going only a quarter of the way, then down , then across , and so on, you'll again travel towards . If you repeated this to infinity, your path would eventually tend towards the hypotenuse, but the distance you travel would still be according to the pattern I described above, when it should tend towards .
I wondered if anyone had any ideas about that. My friend was really really into maths, so it might be a famous problem that people have thought about, but I can't seem to find any information about it.