![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowMan, I love it when this happens.
Let me back up. I've redrafted the section of Taxonomy II on first-order isolated classes, to my satisfaction, and today I began writing the section on higher-order classes. I wrote the boring part, explaining why all but one of them were uninteresting. Then I began on the one that isn't.
This is a second-order class of quadrilaterals, that I've dubbed "Newton-Steiner quadrilaterals". (They include parallelograms, kites, isosceles trapezoids, and quadrilaterals which are simultaneously equidiagonal and orthodiagonal, among others.) I've known about them for a few years, but haven't been able to make much headway with them. I had one abstruse but fairly simple (no, that's not a contradiction) characterization, and, applying my methods, I'd come up with two or three rather ugly characterizations. This was a while back, and I don't remember why they're right; they're just in my notes.
But a day or two ago I got another idea. I wrestled with it in my head, and it seemed sound, but until today I hadn't been able to do the calculations that verify that it works. Today, finally, I plugged the formulas into Mathematica and started crunching symbols; and yeah, the algebra works. But what does it mean *geometrically*? (Stoutfellow's First Law: Algebra produces knowledge, but geometry produces *understanding*.) I stared at the equation. I pushed the symbols around. I did some trigonometry, and got something that looked halfway to being nice. I stared at it some more.
Suddenly it hit me, the right way to look at it. It means that *these* two triangles have the same area but opposite orientation; or, equally well, that these *other* two triangles do the same. I checked out a couple of examples in my head, and they worked.
I've been doing a virtual Snoopy dance all afternoon, and I bought a pint of Ben & Jerry's New York Super Fudge Chunk on the way home. Tomorrow's supposed to be cold again, and I don't have to go onto campus, so I won't - but I *will* be working on that paper!
(I'll explain some of this later, but first I'll have to tell you the Right Way to Think About Quadrilaterals, and that will have to wait.)
Let me back up. I've redrafted the section of Taxonomy II on first-order isolated classes, to my satisfaction, and today I began writing the section on higher-order classes. I wrote the boring part, explaining why all but one of them were uninteresting. Then I began on the one that isn't.
This is a second-order class of quadrilaterals, that I've dubbed "Newton-Steiner quadrilaterals". (They include parallelograms, kites, isosceles trapezoids, and quadrilaterals which are simultaneously equidiagonal and orthodiagonal, among others.) I've known about them for a few years, but haven't been able to make much headway with them. I had one abstruse but fairly simple (no, that's not a contradiction) characterization, and, applying my methods, I'd come up with two or three rather ugly characterizations. This was a while back, and I don't remember why they're right; they're just in my notes.
But a day or two ago I got another idea. I wrestled with it in my head, and it seemed sound, but until today I hadn't been able to do the calculations that verify that it works. Today, finally, I plugged the formulas into Mathematica and started crunching symbols; and yeah, the algebra works. But what does it mean *geometrically*? (Stoutfellow's First Law: Algebra produces knowledge, but geometry produces *understanding*.) I stared at the equation. I pushed the symbols around. I did some trigonometry, and got something that looked halfway to being nice. I stared at it some more.
Suddenly it hit me, the right way to look at it. It means that *these* two triangles have the same area but opposite orientation; or, equally well, that these *other* two triangles do the same. I checked out a couple of examples in my head, and they worked.
I've been doing a virtual Snoopy dance all afternoon, and I bought a pint of Ben & Jerry's New York Super Fudge Chunk on the way home. Tomorrow's supposed to be cold again, and I don't have to go onto campus, so I won't - but I *will* be working on that paper!
(I'll explain some of this later, but first I'll have to tell you the Right Way to Think About Quadrilaterals, and that will have to wait.)
no subject
Date: 2019-02-14 09:40 pm (UTC)no subject
Date: 2019-02-14 09:44 pm (UTC)