Math Satori: Generalized Quadrangle

One thing I came across in various articles about Laguerre planes was references to their "well-known" relationship with the generalized quadrangle. I eventually decided to try to hunt this down and use it for my paper, and it's been a very hard slog, causing me to have to look up term after term after term and try to understand all kinds of things.

The first thing I had to try to understand was what the hell the generalized quadrangle is. I mean, I know what a quadrangle is - a four-sided figure (like a rectangle, square, diamond, etc.) - but there's no obvious relationship between that and a Laguerre plane.

So, it turns out that a generalized quadrangle is a kind of incidence structure, like a projective plane. An incidence structure is something that has two sets - like points and lines - and a relation describing which ones are on or contain each other. So, for instance, if your whole structure consisted of one line, m, with three points, A, B, and C, on it, and a fourth point, D, not on it, your structure would be something like this:

P: {A, B, C, D}
L: {m}
I: {(A, m), (B, m), (C, m)}

where P is the set of points, L is the set of lines, and I is the set of pairs of points and lines that are on each other.

The Euclidean plane is also an incidence structure, but it has an infinite number of points, so it looks more like:

P: all pairs (x, y) of real numbers
L: all pairs (m, b) of real numbers, plus vertical lines that just have an x-intercept (a)
I: a point (x, y) is on a line (m, b) if y = mx + b; also, a point is on a vertical line (a) if its x-coordinate is a

So, the generalized quadrangle is an incidence structure that satisfies these axioms:

1. Two points can't have more than one line in common. (Note: they can have no lines in common, unlike in Euclidean space, where any two points determine a line.)
2. If you have a point A and a point B that aren't on the same line, plus a line m that B is on, there is a unique point C and line n such that A is on n and C is on both n and m.

There will also be some other axioms to make sure the generalized quadrangle is big enough to be worth dealing with. (Under most systems, for instance, a quadrangle is not a generalized quadrangle.)

Now you may have gotten this right away, but I got these axioms after working through a giant example of a generalized quadrangle, and one thing I couldn't see was why it was called a generalized quadrangle. What did it have to do with quadrangles?

In pursuit of this answer, I came across a couple of places that commented that a projective plane is, of course, a generalized triangle. What? I never heard this before. What the...ohhhhh.

I got it.

A projective plane is a generalized triangle because any two points determine exactly one line, and every two lines cross at exactly one point. That's just like a triangle. (Go ahead, draw one and see, if you need to.)

And a generalized quadrangle has the same kind of relationship as a quadrangle:

1. Lines don't cross more than once.
2. If you pick opposite corners, plus a side, you can get from one corner to the other exactly one way, and that trip involves one additional side and one additional point.

Wow.

This may seem kind of trivial, I don't know, but when I got it, I was ridiculously excited. I wanted to race around and explain it to everyone.