Hand and Bush

The thing about group algebras that I talked about a while ago does, in fact, have implications for my research.

It ties together the two big pieces of machinery - the stuff that I started off with, which appears in Taxonomy I, and the newer stuff that I'm introducing in Taxonomy III. They're two manifestations of the same thing, and so *of course* they interact in interesting ways.

It also opens the door to an enormous generalization, to "G-gons", where G is a finite abelian group. Never mind what that means; the integers mod n form a finite abelian group, called Zn, and an n-sided polygon, under my Taxonomy I definition, is precisely a Zn-gon - but there are other possibilities!

This looks like it might be fascinating, but *I've got papers to write* - papers that I *know* contain interesting results. Bird in the hand...

I'm going to have to divide my time, this summer. The new stuff may be a will-o-wisp, but if not, it's *really neat*. Birds in the bush...

:moan:

Something I Should Have Said On March 14

In high school geometry, people learn formulas. "The area of a circle is pi r squared". "The circumference of a circle is 2 pi r." As far as I know, they don't learn the real point, the magic of Euclidean geometry.

Here's the magic: the ratio between the area of a circle and the square of its radius is the same as the ratio between its circumference and its diameter, and that ratio is the same for all circles.

That's it. That's what those formulas are about. In a non-Euclidean space, they are false - false in all clauses.

I'll admit that I didn't see that that was the point until I was middle-aged, and I speak as a professional mathematician; it's no wonder layfolk never see it.

How shall they know, if they are not told?