A good ramp up into abstract algebra is the idea of a group. Groups show up everywhere in mathematics, and getting a feel for working with them really helps you learn about other algebraic notions.
There are a number of ways to think about groups, but for now I’ll stick with a very concrete, hands-on approach. This is the sort of thing you’d run into in a first undergraduate course in abstract (or “modern”) algebra.
So, a group is basically a set (a collection of elements) with some notion of composition defined which satisfies certain rules. That is, given two elements and of a group, there’s a way to stick them together to give a new element ab of the group. Then there are the
- Axioms of Group Theory
- Composition is associative. That is, if we have three elements , , and , the two elements and are equal.
- There is an identity. That is, there is an element (usually denoted ) so that .
- Every element has an inverse. That is, for every element there is another element so that .
That’s all well and good, but if this is the first time thinking about an algebraic structure like this it doesn’t really tell you anything. What you need (after the jump) are a few
This semester I’m sitting in on Jayadev Athreya’s course on billiards. “Billiards?” I hear you cry, “The game like pool but with all the red balls?” No, that’s snooker. Besides, the course is on (not surprisingly) a mathematical model inspired by balls bouncing around on a table.
When playing mathematical billiards, we place a ball down on a polygonal table and send it off in some direction. When it hits the edge of the table it rebounds, making the same angle as it leaves as it did when it came in. We simplify a bit by assuming that the ball isn’t spinning, has no friction with the table, and so travels at a constant speed in a straight line between bounces.
So let’s start with a square table. How do we determine how the path behaves as time goes on? Imagine the ball approaching a side of the table. Instead of the ball reflecting off the edge, let’s reflect the whole table through the edge and let the ball continue on its straight-line path. Whenever the ball’s about to hit an edge, we have a reflection of the table ready for it.
But we really don’t need all that many reflections. In fact, four will do it: the first table, a horizontal reflection, a vertical one, and one reflected in both directions. We can put these four squares together into one bigger square. Instead of reflecting the ball at an edge, we can just let it wrap around to the other side of the table travelling in the same direction, just like a game of Asteroids. That means we’re really just rolling a ball across the surface of a torus.
Now the whole picture is pretty clear: by Weyl’s criterion, if the tangent of the angle between the path and an edge of the table is rational the path eventually closes back up and runs over itself again. If not, the path covers every point on the table equally. By this I mean that if the whole table has area , then given a section of the table of area , the ball spends of its time in that section.
There’s a lot of interesting material here, and a lot of it can be broken down to bite-sized chunks, tying into all sorts of other areas of mathematics. I’m starting a category (in the WordPress sense) so if you’re interested, there will be plenty more.