I had a little progress on my thesis work, recently. Essentially, I was able to prove what form a composition of transformations would take in the most general case. I had a hunch (and a hope) that it would be the simplest thing I could think of: given two transformations defined by two numbers, the composition of them should be defined by the product of those numbers. And indeed, that’s the way it is, in the simplest case.
This is one of the reasons I like math: things sometimes work out like they’re supposed to. If you study biology, it seems like everything is a hopeless mess; any process in the body is affected by every other process, so you can barely ever get a clear look. In math, there’s this magical way that things end up being surprisingly simple when you look at them the right way. That’s what seems to be happening with my work: a complicated transformation involving geometric structures connected via an abstract algebraic process and the whole thing boils down to multiplication.
For the moment. I have a feeling the subsequent cases will be a bit rougher.