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Dark/Light
Jun. 21st, 2019 09:48 amSomeone once said that the early stages of mathematical research are like blundering around in a pitch-dark room, bumping into things, trying to visualize them and remember where they are, until that blessed moment when you find a light-switch.
That's where I am with "G-gons". I have very little idea of how they work, as yet. I'm doing experiments, working out examples, looking for patterns. Patterns lead to conjectures, conjectures lead to theorems...
Yesterday, though, I found a small side-room's light switch. It doesn't illuminate anything outside the room, but I can now see what's there, at least.
I mentioned that, in some cases, a lot of the machinery I've developed for polygons doesn't carry over. There's an infinite sequence of G's - the 4-group V is the first - where some of the machinery simply collapses. My previous work features a fundamental distinction between "primary" and "secondary" classes; for these choices of G, there aren't any secondary classes, and so a lot of my work becomes moot. In addition, the special roles played by "central" and "quasi-central" transforms dissipate: every transform is central, and all that line of reasoning becomes useless.
But. I just got my first theorem, and it applies to precisely those misbehaving G's. I know what the first-order simple classes are for those groups; it's a neat generalization of what they are for V.
Lux fuit.
That's where I am with "G-gons". I have very little idea of how they work, as yet. I'm doing experiments, working out examples, looking for patterns. Patterns lead to conjectures, conjectures lead to theorems...
Yesterday, though, I found a small side-room's light switch. It doesn't illuminate anything outside the room, but I can now see what's there, at least.
I mentioned that, in some cases, a lot of the machinery I've developed for polygons doesn't carry over. There's an infinite sequence of G's - the 4-group V is the first - where some of the machinery simply collapses. My previous work features a fundamental distinction between "primary" and "secondary" classes; for these choices of G, there aren't any secondary classes, and so a lot of my work becomes moot. In addition, the special roles played by "central" and "quasi-central" transforms dissipate: every transform is central, and all that line of reasoning becomes useless.
But. I just got my first theorem, and it applies to precisely those misbehaving G's. I know what the first-order simple classes are for those groups; it's a neat generalization of what they are for V.
Lux fuit.
