A Little Math

One of my courses this semester is Foundations of Geometry, which sounds like it would be easy, but so far has been pretty interesting, and has forced me to be more algebraically clever than usual. (Algebra, in a geometry class? Say it isn't so!)

When you have geometry in high school, it's usually Euclidean geometry, in two senses. The most important way in which it's Euclidean (in my view) is that it uses axioms and proofs to develop geometric ideas. Euclid totally invented this approach to math. Some people (like me) love this and other people (like Mosch) think it's a big waste of time. The secondary sense in which it's Euclidean is that it uses the specific axioms of Euclid. If you change one of Euclid's axioms (the parallel postulate), you can get some different interesting results.

My class is currently proceeding down two lines of work - some straight-up Euclidean stuff, usually presented as puzzles at the beginning of every class period, which we solve and then do some proofs about, and then "analytic" geometry.

Analytic geometry is where you say, "Why is Geometry the only part of math where we're still doing this weird Euclid-type stuff? Let's do everything with algebra and calculus instead!" And so it goes. Our textbook (a slender volume called "Modern Geometries: Non-Euclidean, Projective, and Discrete," by Michael Henle) uses this approach.

In analytic geometry, at least as presented in this course, you represent 2D shapes as coordinates in the complex plane. (This is basically like the regular cartesian plane where you have x, y coordinates, except that in this case the y axis represents the imaginary part of a complex number, so every point in the complex plane is actually just one number of the form x + iy.) Then you use regular math to prove stuff about them.

For instance, one thing you do a lot of in geometry is prove that things are congruent. "Congruent" basically means they are the same size and shape. Euclid's version of congruence is that two shapes are congruent if you can pick one up and lay it on the other and everything lines up. This makes sense, right? If you have two triangles on two pieces of paper, you can literally pick up one sheet, put it over the other, rotate it and move it around, and see if the triangles are the same or different.

In analytic geometry, you prove two things are congruent by proving that there is a one-to-one function (of an allowable type, like a rotation) that takes the set of points contained in one shape and transforms them to the set of points contained in the other shape. This is basically the same thing as what Euclid did, except that it's all algebra instead of being a physical maneuver.

These functions that change one set of points to another set of points are called "transformations" and the ones that Euclid would allow are basically rotation (where you turn something around), reflection (turning the piece of paper upside down, if you had the shapes on paper), and translation, which just means moving things up or down, side to side, or diagonally. If you think about how you'd line up shapes on two pieces of paper, those are the basic moves - you rotate the paper, flip it over, move it around, or some combination of those things. If you can change one shape into another by those types of maneuvers, then the two shapes are congruent.

So...that's about as far into analytic geometry as my class has gotten at this point. I could say more about it, but it would go into weirder math, so I'll refrain :-)