Quite some time ago, a friend asked me what would happen if we tried to construct a torus by gluing all 4 sides of a sheet of paper together, instead of first one pair then the other. Didn’t the math have to specify first one pair then the other?
One reason I’ve been hesitating over this post is that it doesn’t seem to be “real” mathematics – though any number of people might howl that PCA / FA isn’t “real” mathematics either. This is just a small drawing that I cobbled together to show that the homeomorphism between a circle and a line segment with endpoints identified… well, it doesn’t have to correspond to a physical process. (Why don’t I refer to “the glued line”?)
“… algebra provides rigor while geometry provides intuition.”
from the preface to “A Singular Introduction to Commutative Algebra” by Greuel & Pfister
It helped me to go back and read my original comment when I acklowledged Jim’s question to this post. I see that I did not understand that what matters to the formalism, the algebra, is before and after; what matters to the geometry is between or during. We reconcile them by permitting some things in the geometric visualization that we would not permit in the formal algebra, if the algebra even formalized the process: points passing thru points; or even some tearing, if when we reglue it we restore it rather than take the opportunity to change it.
That’s what bloch says, on p.57, discussing the physical process of getting from the knotted torus
to the regular torus.
In contrast, the sphere movie says going thru is ok, but tearing is not. (That’s the same link cited in my original comment.)
Keep it simple. Jim’s comment was that deforming and gluing a flat sheet to form a torus requires another dimension, namely time. To be specific, Jim considered 3 processes:
- glue the left and right sides together, then glue the two circles.
- glue the top and bottom sides together, then the two circles.
- glue both pairs of sides together simultaneously.
He cannot see that the third process generates a torus. Well, neither can I.
I’d like to think that it does, but it doesn’t have to.
The formal mathematics doesn’t actually deal with the deformation process we’re imagining; rather it shows “before” and “after”. Don’t get me wrong: the formal math is motivated by the deformation process, and if the formal math hadn’t led to results which were by-and-large plausible, we would have changed the math.
The mathematics requires that a function describing “before” and “after” be a homeomorphism: 1-1 and both the function and its inverse are continuous. But this isn’t a description of “during”. We describe the result of gluing, not the process of gluing.
Let me show you a much simpler drawing than a torus, simpler even than a cylinder. I start with a line segment of the x-axis, and I want to transform it into a circle. If I imagine that I am wrapping the line onto the circle – wrapping a strip of paper mache’ onto a frame – then I could come up with the following mapping.
It starts at the origin and moves CWW (counter-clockwise); t runs from 0 to .
Now, draw lines from “before” points on the line to “after” points on the circle. Let’s do it first for just the leftmost quarter of the line segment, :
add the next quarter, :
We see a problem as we move toward the top. As one example of many, the line L to the top of the circle, mapping the point on the line to the point (0,2) on the circle, passes through the circle. That is, it passes thru points that have already been mapped to the circle. If we imagine that these lines had been traced out at constant speed, but stopping when they reach their destination on the circle, then the line L hits and goes thru a point on the circle.
The problem is even clearer when we add mappings from the last half of the line segment, since all the lines which are mapped to the left of the circle must pass thru the right of the circle, and all of those points have already been “installed”, as it were. Let’s see this with the 3rd quarter of the line segment :
This process was a far cry from “wrapping” the line segment onto the circle. But such processes have nothing to do with the homeomorphism between the glued line and the circle.
The homeomorphism is a relationship between the glued line and the circle; not a relationship from the glued line to the circle. There are many processes from the glued line to the circle, but not all of them are nice. And this is one that isn’t nice.