New Method

A one-edge solution

As you have seen, all my methods of trisection have depended on using a second edge of an ordinary ruler, thus dispensing with the need to make marks on it. Recently I started wondering if it would be possible to use only one edge of an unmarked ruler to trisect. This being impossible, of course, without cheating, I realized that the only possibility would almost certainly hinge on violating the Euclidean demand that the straightedge be of infinite length. In other words, the length of the ruler would have to be usefully employed. I still wasn't at all sure that it could be done, but it seemed like a worthwhile intellectual challenge. If I could do it, I would be able to show that it is indeed possible — with some clever cheating again — to not only trisect with just a compass and an ordinary unmarked ruler, but to use only one edge of it, which on the face of it would bring us closer to meeting the terms of the ancient challenge in practical terms because it would use a simple straightedge, just not one of infinite length (which does not in any case exist — and cannot).

I have indeed now found a simple and elegant way to do exactly that. As I have already noted on the first page of this site, my method will be announced here as soon as it has been published in an academic journal, an event that is already being discussed with the editor of a respected British peer-reviewed journal. As the wheels of publication often grind slowly, it may be some time before I can update this page.