I was walking home the other night after a less than draining day at the office (see numerous previous posts) and my mind was meandering. It started as I approached the area of common land that I have been walking alongside for the last few months. Walking on the inside path cuts off the corner that the pavement goes round. So far so good, everyone knows that cutting a corner reduces the distance that you travel. However, now that the weather is improving, I have a second opportunity to cut a corner, by going across the grass, which is now dry and not muddy. After a lot of mental gymnastics, I figured out that the way to minimise the distance travelled is to walk along the equivalent of the hypoteneuse, assuming that the corner you are cutting is a right angle. If you cannot do that, the next best thing is to start cutting across as soon as possible, and walk to the point closest to the end point of the triangle of ground that you are cutting off.
Now, I know you are asking yourselves now, so what? This is really simple geometry. Everyone can figure this out, in fact it is almost instinctive. I agree. I just did that to lull you into a false sense of superiority. Here comes the bit I just do not understand, and if anyone can explain it to me, I would be really grateful.
Imagine that, instead of walking along that hypoteneuse, you walk a zig-zag path, making a series of triangles against the hypoteneuse. If you add up all the distance walked, you will find that you have travelled exactly the same amount of distance as if you had walked the other two edges of the triangle. For example, take the classic triangle with sides 3 and 4 units, with a hypoteneuse of 5 units. If you walk the two sides, you have travelled 7 units. This, incidentally, is the equivalent of walking one zig-zag along the hypoteneuse. Now walk two zig-zags. You will walk 1.5 units down, 2 across, 1.5 down, and finally 2 across. That adds up to 7 units, as expected. Do four zig-zags and the path distance still adds up to 7 units – four lots of 0.75 down followed by 1 across. This is all pretty mundane, but now comes the tricky bit.
In my youth I was foolish enough to do some advanced maths, including calculus. I know there must be a mathematician reading this who can see where I am going with this, so please leave a comment explaining! Right, let us imagine that I now take a very large number of zig-zags to walk that hypoteneuse, so many in fact that you move such a small distance that you effectively do not leave the hypoteneuse. How far do you end up walking? Would it be 7 units or 5 units? As far as I know, this is a calculus problem, but in a pragmatic sense I cannot see which result is correct, or if there is a path length at which the result flips from one to the other. Perhaps I am just asking the wrong question, or my model is wrong. Any help deeply appreciated.
That, dear readers, is why I am not a mathematician.