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Inevitable
Jan. 24th, 2019 03:36 pmOne of the things that can make a class of polygons interesting is multiple characterizations - properties of which one can say, "A polygon is in this class if and only if it has such-and-such property." If you get enough of them, you can pile them into a single theorem: "For any polygon P, the following are equivalent. Item 1, Item 2, Item 3,...".
The paper I'm currently writing has to do with what I call "isolated" classes. The thing about isolated classes is that there are some simple techniques for generating characterizations almost without effort. I have to check that they aren't tautologies - that they aren't properties that all polygons with that many sides have - but that's easy to do.
I'm writing up the section on isolated classes of quadrilaterals. I'm down to the last two of them. I quickly wrote down three characterizations of the first one, but wasn't very happy with them; the fact that they are equivalent is trivially easy to prove. I started looking for more exotic results... and my techniques suddenly failed me. I kept coming up with possibilities, but every one of them was either a slight variant of one I already had, or was tautologous. After more than an hour spinning my wheels and doodling on Sketchpad, I finally threw up my hands and headed for home.
As I was waiting for the bus, the inevitable happened. A new line of attack occurred to me; another, very different characterization appeared, and then another, and a general idea of how to find more. For the second one, I had to wrestle for a while; without pen or computer, I had to try to visualize what was going on. I finally convinced myself that it made sense, both algebraically and geometrically.
Fortunately, I can access my office computer from home, and I'll probably write up some of the new stuff later today. But why couldn't it have come to me an hour earlier?
The paper I'm currently writing has to do with what I call "isolated" classes. The thing about isolated classes is that there are some simple techniques for generating characterizations almost without effort. I have to check that they aren't tautologies - that they aren't properties that all polygons with that many sides have - but that's easy to do.
I'm writing up the section on isolated classes of quadrilaterals. I'm down to the last two of them. I quickly wrote down three characterizations of the first one, but wasn't very happy with them; the fact that they are equivalent is trivially easy to prove. I started looking for more exotic results... and my techniques suddenly failed me. I kept coming up with possibilities, but every one of them was either a slight variant of one I already had, or was tautologous. After more than an hour spinning my wheels and doodling on Sketchpad, I finally threw up my hands and headed for home.
As I was waiting for the bus, the inevitable happened. A new line of attack occurred to me; another, very different characterization appeared, and then another, and a general idea of how to find more. For the second one, I had to wrestle for a while; without pen or computer, I had to try to visualize what was going on. I finally convinced myself that it made sense, both algebraically and geometrically.
Fortunately, I can access my office computer from home, and I'll probably write up some of the new stuff later today. But why couldn't it have come to me an hour earlier?
