Wednesday, August 4, 2010
Wednesday Math, Vol. 123: Metric spaces
What is the distance from Point 1 to Point 2? As word problems go, this is one of the most basic, and luckily for us, there's usually a picture to go along with it. In this picture, we aren't told what unit of measurement is being used, but the drawing makes it clear that we are in three dimensions. The red line is a hypotenuse of a right triangle and the blue line is as well, so if we call Point 1 (0, 0, 0) and Point 2 (x, y, z), using the Pythagorean Theorem twice gives us the distance is the square root of x² + y² + z².
Distance problems are sometimes not so straightforward. Back two years ago, I wrote a post about distance on the sphere, which involves arcs on a great circle instead of the Pythagorean Theorem. There are plenty more examples and new methods and formulas to use to determine what we mean by distance.
Consider a grid of train lines like the one on the left. Now, we are limited to just a finite number of points, the different train stations, but if the entire system has two way travel, it can be a metric space, the generalized version of the idea of distance defined on a set of points.
A metric space is defined by a very general set of rules, which I list here.
1. For any two different points A and B in our metric space, there is a positive number d(A, B) which we will call the distance.
2. d(A, B) = d(B, A), which is to say the return trip is the same length as the original trip. This is why I brought up that two-way travel on all paths is required. If there are one way streets, the distance from A to B might be a single city block, but from B to A might take a trip around the block.
3. For every point A, d(A, A) = 0. Combined with rule 2, this says that if d(A, B) = 0, then A and B must be the same point.
4. The triangle inequality. This says there are no "shortcuts". If I travel from A to B to C, the distance traveled must be greater than the distance from A to C, which means B forced me to take a detour, or it can be equal to the distance from A to C, which means B was on the path from A to C.
Ways of measuring distance in some systems can become very convoluted or very strangely unintuitive, but if the system follows these four simple rules, it's a metric space.
One of my favorite metric spaces is the discrete metric. It follows two simple guidelines for defining a distance.
1. d(A, A) = 0 for all points A in our set.
2. d(A, B) = 1 for any two distinct points A and B in our set.
It's like saying there are two distances, "right here" being distance 0 and "not right here" being distance 1. The first three rules of a metric space are satisfied by the definition from the guidelines. Proving the Triangle Inequality is fairly straightforward if you understand the rules, and will be left as an exercise for the reader.
There's a lot of theorems dealing with metric spaces and how they arise in math. If you have two different ways of measuring distance on the same set of points, there are ways to combine metric spaces to create new metric spaces. For example, the maximum distance between any two points using two different distance systems must also be a metric space, but not the minimum distance of two systems. When I was an undergrad, I proved that the minimum of the discrete metric and any other metric was in fact a metric, and I gave this new metric the name the Provincial Metric. One way to think of it is that anyone at point A knows exact distances to nearby points, but past a certain distance, all other points are just "not particularly close". I was kind of proud of figuring this out on my own when I was younger. I've seen examples of it in textbooks printed after I got my undergraduate degree, but none of them call it the Provincial Metric, which I will say with no modesty is the best name for the idea.