RW: Dr. Brooks, could you explain or give examples of geometrical shapes derived from a sphere?
RB: Let me give you an example of the torus, which resembles a doughnut. This was part of a lecture I gave in my class, which dealt with the Classification of Surfaces. This is one of the great theorems of topology. To set up the statement of the theorem, take a sphere(see Figure 3). Now take a torus(see Figure 4). Then take a two-holed torus(see Figure 5). And now down below you have a projective space(see Figure 6). It’s not oriented, so you can’t draw it properly. And then take the Klein bottle, which you also can’t draw(see Figure 7).
RW: Right. The Klein bottle is the surface that continues into itself without end, somewhat like a three-dimensional Mobius strip.
RB: Yes. Let’s return to the projective space(see Figure 8). We’ll put a little handle on it(see Figure 9). You can take other spaces, and put more handles on them.
RW: So you’re taking the basic sphere, and adding holes.
RB: Well, think of it as adding handles. A torus is a sphere with one handle, and the two-holed torus is a sphere with two handles. To understand what I mean by handle, just think of what’s attached to your luggage. So if you think of your luggage as a sphere, and attach a handle to it, you get Figure 10a. And then if you start shrinking the sphere down and pull the handle, you get Figure 10b. Eventually, your shrinking and pulling will result in Figure 10c. And so the shape in Figure 11(a-c) is the same as pulling the two handles on the sphere. And then you take the real projective plane(the space of all lines going through the origin. It is obtained by identifying opposite points on a sphere. It is also called the cross cap.) and the Klein bottle, and you put handles on them. By taking a square, and folding the appropriate ends such that the arrows are lined up and pointing the same way, you can obtain the shapes we have been discussing: torus, Klein bottle, sphere, and real projective plane(see Figures 12a-12d).
First, I will define a surface as an object such that any sufficiently small piece of it can lie in the plane. Second, I will assume that the surface has no boundary. For instance, if you puncture a sphere, it goes out like a balloon, and you can imagine it continuing like an infinite plane. The third thing that I’m going to insist on is that the surfaces are compact, which means whenever you have infinitely many points, they have to pile up.
RW: So by compact we’re placing a limit?
RB: Yes. Then the question is, “What are all the possible surfaces, if you call two surfaces the same if you can obtain one from the other by deforming them through stretching?”(See Figures 13-19). And the theorem states that these shapes form a complete list. Okay, take the sphere, and a number of handles on it. Take the projective plane, add a number of handles on it. Take the Klein bottle, and add a number of handles, and that will form all possible surfaces. Okay, in the terms that I specified in this theorem are all the two-dimensional shapes which don’t have any boundary and are compact. This is it. It’s beautiful, you know. I don’t know if you count this as an abstract theorem.
RW: Well, it’s a generalization, which is an abstraction of sorts; moving from a specific example to some kind of rule.
RB: From that point it is an abstraction. You go to the proof, and, in step one, you list all the parts that such a surface might be built out of. There are seven parts.
RW: It’s interesting when we look at these shapes that some of them look very familiar. The upside-down version of the Y-shape looks like a pair of pants on a store dummy.
RB: Actually, the technical term for this shape is “pairs of pants.”
RW: So your point is that there’s something is that there’s something very concrete about all of this. We can recognize some things as ordinary objects. Yet, it’s also true that we are aiming for certain generalizations about these objects. This seems to me to be somewhat abstract.
RB: I think, in terms of the metaphor you used about looking into the soul of an object, that the soul needn’t be some abstract thing. It could be made out of bits and pieces of these shapes. On the one hand, it’s abstract in the sense that you’re looking into the essence of things. On the other hand, it’s concrete; you can sit down and play with these shapes with your children.