As I mentioned the other day, I recently acquired a neat little book called Experiments in Topology by Stephen Barr, dating from 1964, though you can get a paperback reprint from Dover Books. Since I’m still at my parents’ house for another two days, I don’t have access to all my other books or (more importantly) the graphics programs I use, so I won’t attempt to liveblog my reading of the book. However, in the spirit of my earlier posts on Möbius and topology, I borrowed some scissors and raided the local paper supply to illustrate in photos what I can’t diagram.
To recap: topology is a branch of mathematics dealing with features of surfaces that aren’t related to their basic shapes or sizes. Topology allows you to squish and stretch shapes, but doesn’t allow tearing holes, gluing edges together, or closing up existing holes. Thus, a donut (one hole) is not topologically equivalent to a sphere (no holes), but cube is topologically equivalent to a sphere, a donut is topologically equivalent to a coffee cup, and so forth. Brass knuckles, having four holes, aren’t equivalent to any of those (and I mention them just because it’s probably the only time I’ll ever have a reason to discuss brass knuckles on this blog).
Just as shapes can be stretched and compressed, lines drawn on a surface can be stretched and moved around. However, just like you can’t make new holes, lines can’t be cut and loose ends can’t be joined. A circle is equivalent to a square, a straight line is equivalent to an S-shape, but a straight line can’t be turned into a circle. A figure-eight is not equivalent to either, since the line intersects itself in the middle. I’ll return to some cool stuff you can do with curves and intersections in a later post, but for now let’s think about simple curves on some of the surfaces I’ve already discussed.
Specifically, if you were confined to a two-dimensional surface, how might you be able to tell if there’s a hole? How might you be able to tell a cylinder from a sphere, a torus from a Möbius strip,
a green field from a cold steel rail? Again, we’re using topology, so a “hole” isn’t a literal hole you could fall into: it’s a property of the surface itself. Let’s start on a plain piece of paper with no folds, rips, or holes. Any closed curve that doesn’t cross itself (triangle, square, etc.) will be equivalent to a circle topologically; its location on the paper doesn’t matter, since every place on the paper is just like every other if we pretend the paper has no edges. We say that the paper is simply connected: any two points can be connected with a single type of curve without any kind of gymnastics. Another equivalent way to think of it is that any closed curve can be shrunk down to a dot, since there are no obstacles to doing so.
Now let’s cut a hole in the paper, as shown. Suddenly we have a special kind of closed curve: one that surrounds the hole. The two curves in the picture can’t be turned into each other without cutting, since any curve intersecting the hole won’t be closed: it will have two ends, even if the hole is really small. This space is not simply connected: the hole is a kind of topological defect (though I dislike the connotations of “defect”, since this is merely a property of the space, not a flaw). You can’t shrink the curve surrounding the hole down to a dot.
Two curves on a normal piece of paper like this (with or without a hole) can meet at a single point, but they can’t intersect at only one point. One may enclose the other, but if they intersect, they must cross twice. This also is true even if one curve isn’t closed, as long as it doesn’t have end points.
A cylinder is a two-dimensional surface that curves back on itself in one direction, so you can draw a line around the circumference that connects with itself, yet isn’t a closed curve in the same way a circle is. Thus, cylinders are not simply connected. In fact, a cylinder is very like a flat piece of paper with a hole in it, since there are the same two types of curves, with the “inside” of the cylinder acting like the hole.
If you cut along the circumferential line, the cylinder becomes two cylinders. This type of line, called a Jordan line for the mathematician who characterized it, divides the cylinder into two regions. Cutting along a more “normal” type of closed curve makes a hole, changing the type of surface. (So, dear readers: is the Equator a Jordan line or not? How about the Arctic Circle?)
A torus can be made by connecting the ends of the cylinder together. (I’ll post my plans to make your own torus next week, if I remember to do so.) Since it’s based on a cylinder, the torus is not simply connected either, but it’s even more fun: there are two types of circumference lines that can’t be deformed into a single point. Unlike the other lines on cylinders and flat papers, the circumferential lines intersect at one point, as shown in the picture on the left. Life on a torus is life in the world of the classic video game Asteroids: leaving one side of the screen (following one of the circumferential lines) always brings you back on the opposite side.
Finally, what about a Möbius strip? Just like a cylinder, it isn’t simply connected: you can draw a line along the circumference that can’t be shrunk to a single point. However, this line is not a Jordan line: if you cut along the circumference, you don’t split the strip into two! (Here’s your homework for today: is the new strip you create another Möbius strip or not?)
We haven’t fully answered the question of how to distinguish whether there is a hole in a surface, but we can see how it can be done. By tracing multiple closed paths and seeing if they intersect in one point or two, and whether they divide the space into two or not, we have an inkling of the topological character of our surface.
Now for some physics, which I will definitely elaborate in a follow-up post. An electric charge or a mass is akin to a hole in a surface: going to three dimensions instead of two, a closed curve becomes a closed surface. A surface enclosing a charge or mass is not the same as a surface enclosing empty space, and they can’t be deformed into each other without tearing holes. The strength of the electric or gravitational force is then determined by the size of the surface containing the charge or mass. (Those with a bit of physics from high school or college may recognize the origins of Gauss’ law in what I’m writing here.) Farther afield, the central singularity of a black hole is a true topological defect, a place where the normal laws of physics break down. We can even talk about topology in the context of simple pendulum dynamics: in phase space, the points of equilibrium, where a pendulum comes to rest, is a topologically important spot where trajectories end.
I will continue to blog my way through Experiments in Topology, tying what I learn in with physics. Hopefully you all will have as much fun as I’m having.