Parallel Convergence

The Full Proof!

I think this blip disproves Euclid's Fifth Postulate, the one that, in basic terms, says that parallel lines never meet. Ilkley Moor has a history steeped in magic and mystery but this kind of non-Euclidean geometry is normally the preserve of cosmological black holes! More seriously, though, I do wonder if the remarkable perceived parallelism of these drystone walls is the result of accident or design. This is a companion blip to Divergence, taken from just a few yards away and on the other side of this wall.

It occurred to me today that I've really no idea how old these walls are. I kind of assume that they must date back over a century but I can't be sure of that. I do know that it would cost a small fortune to construct such a wall today, so they must certainly date back to a time when skilled labour was very much cheaper in real terms. I doubt much construction would have been achieved on a day like today however. Although the sun shone brightly the overnight gale continued to blow unabated. It finally felt like winter up on the moor this afternoon. I was actually intending to go for a decent bike ride, but I wimped out because of that cold wind. It's easier to keep warm while running. I'm trying to build some foundation endurance fitness for next year so I'm hoping today's steady run will have helped a bit in that respect.

I'm afraid I continue to be a rather hopeless community member here. I was fully expecting to be able to spend some time browsing today but a number of issues came up on Friday when I was away which have consumed far more time than expected. I really, really appreciate so many people stopping by to comment. But for your support I know that this weekend's blips would have been rather mediocre. I may well have not bothered to get out at all yesterday! So thank you, each and every one of you, once more.

Postscript

On closer inspection, the perspective view of these two walls shows the lines to be converging, although somehow our eye adjusts to make them look closer to parallel than they actually are. In Divergence it seems clear that the walls are diverging, although actually not by very much at all if you look closely. There is an illusion here where our mind works to represent the view to us in a way that is more acceptable to our experience.

We can therefore infer that the critical angle lies somewhere between the angles subtended in these two shots. If you look here you can see a satellite photograph of the three walls in question, courtesy of those amazing folk at Google. The key point to grasp in order to work out the angle is that the apparent size of any object decreases in direct proportion to its distance from us. Therefore, in order to maintain the illusion of being the same distance apart, the walls always have to be separated by exactly the same distance that they are removed from us. A simple bit of trigonometry shows that the angle is the inverse tangent of one half - which equals just over 53 degrees.