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stoutfellowI'm really pleased with how the first Taxonomy paper is going. I'm about halfway through the final section.
What I'm enjoying is this. The final section applies the machinery built in the earlier sections to triangles, working out "first-order", "second-order", and "third-order" classes of triangles. But it's not simply a threefold repeat of the same sort of discussion. The first-order classes are "isolated classes", which are special in lots of ways; each isolated class typically has multiple characterizations (i.e., lots of theorems). In the second order, we don't get any isolated classes; instead, we get one-parameter families. That is, in each situation (there are three of them), we have a parameter t, and each value of t corresponds to a particular class. Typically, each triangle belongs to one class in the family, so t is a parameter of the triangle, and we can ask what that parameter says about the triangle. (In one case, it turns out to be equivalent to a known parameter - but it has additional implications.) In the third order, the families are two-parameter, and turn out to yield a "parameter space" for triangles - i.e., knowing both parameters, you know everything interesting about the triangle, and there's a pictorial image of the set of all triangles, within which interesting things can be seen.
The point is that my work on taxonomy of polygons can be used in multiple directions, including some which I didn't foresee when I began this project. (And I haven't even gotten to the really neat stuff - that'll be in the third paper!)
What I'm enjoying is this. The final section applies the machinery built in the earlier sections to triangles, working out "first-order", "second-order", and "third-order" classes of triangles. But it's not simply a threefold repeat of the same sort of discussion. The first-order classes are "isolated classes", which are special in lots of ways; each isolated class typically has multiple characterizations (i.e., lots of theorems). In the second order, we don't get any isolated classes; instead, we get one-parameter families. That is, in each situation (there are three of them), we have a parameter t, and each value of t corresponds to a particular class. Typically, each triangle belongs to one class in the family, so t is a parameter of the triangle, and we can ask what that parameter says about the triangle. (In one case, it turns out to be equivalent to a known parameter - but it has additional implications.) In the third order, the families are two-parameter, and turn out to yield a "parameter space" for triangles - i.e., knowing both parameters, you know everything interesting about the triangle, and there's a pictorial image of the set of all triangles, within which interesting things can be seen.
The point is that my work on taxonomy of polygons can be used in multiple directions, including some which I didn't foresee when I began this project. (And I haven't even gotten to the really neat stuff - that'll be in the third paper!)
