Wednesday, April 08, 2009

The Isomorphic Thrill

The other night, I decided to turn momentarily away from the Polster/Steinke line of thought on Laguerre Planes (covered in my previous post) and look at one of my other sources. The third author I have a lot of writing by, including his dissertation, is Robert Knight. I opened up an article by him...something about bundle forms and "ovoidal Laguerre planes."

My impression of Knight so far, from lightly perusing some of his stuff, is that he's mostly going kind of old school with the Laguerre business. And indeed, in this paper, as background, he presents a set of axioms that I think are pretty close to Laguerre's own.

In the original Laguerre system, instead of circles and points, we have cycles and spears. Cycles (in the original representation) can be either oriented circles or points, and spears are oriented lines. That's my understanding at least.

But when I started jotting down the axioms, I was very excited to find that they were more or less isomorphic to the Polster/Steinke set. I got that pure mathematical thrill as I confirmed that my octahedron "plane" satisfied all of them (treating the points as spheres, the faces as cycles, and with one other additional but obvious rule to account for something not present in the other system).

A phrase I've encountered in this course, but not before, is "up to isomorphism," as in, "There exists a unique field of order 2, up to isomorphism." What it means is that you might be able to make different ones, but they all have the exact same structure even if the elements have different names. And, in a more general sense as well, I've encountered a ton of isomorphism in this class.

The obvious approach to hearing about Laguerre planes, for instance, is to ask "What is a Laguerre plane?" And you expect to get an answer, maybe something like, "Well it's like a normal plane except it curves around," or, "It's the surface of a sphere," or, "They call it a plane but it's really an octahedron." But there just isn't actually an answer like that. What makes something a Laguerre plane is satisfying some set of axioms about Laguerre planes, from which a bunch of theorems will then hold. Anything that satisfies those axioms is a Laguerre plane.

I mean, I had that octahedron, but there's nothing special about that. The Polster book had another finite example with the same number of points and circles; I found the octahedron easier to represent for the blog post. But the two were (clearly) isomorphic.

And that's the way in which the Polster axioms and the Knight axioms are (more or less) isomorphic as well: the same conditions satisfy (or pertain to) both.

As I mentioned, this kind of stuff gives me a real mathematical high when I get it.

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