All that talk about numbering spots on the New York Thruway had a goal, that of establishing how we could set up a coordinate system for the points on a line. It turns out just as easy to do this for a curve, even one a little bit complicated like a branch of the Thruway. About the only constraints we said anything about were that we shouldn’t have branches. Lurking unstated was the idea that we didn’t have loops. For the Thruway that’s nothing exceptional; if we had a traffic circle in the middle of a high-speed limited-access highway we wouldn’t very long have a high-speed highway. Worse, we’d have some point — where the loop crosses itself — that would have two numbers describing its position. We don’t want to face that. But we’ve got this satisfying little system where we can assign unique numbers to all the points on a single line, or even a curve.
The natural follow-up idea is whether we can set up a system where we can describe a point on a surface or even in all of space using the same sort of coordinates scheme. And there’s the obvious answer of how to do it, using two numbers to describe where something is on a surface, since that’s a two-dimensional thing; or three numbers to describe where it is in space, since that’s a three-dimensional thing. So I’m not going to talk about that just now. I want to do something more fun, the kind of thing that could do nicely in late-night conversations in the dorm lounge if undergraduates still have late-night conversations in the dorm lounge.
If we have a long enough thread, or a strand of yarn, or whatever the quite correct term is, we know this can be set up with a coordinate system by marking off distance along that thread. We imagined doing that, more or less, with the numbering system on the Thruway and imagining the straightening out and curving and other moving around of the highway’s center line. As long as we didn’t stretch or compress the strand any, we could spread it out in any shape we liked, and have coordinates for whatever path the strand traces out.
So: if we have the strand running back and forth, close enough to itself, can’t we touch all the points, if not within the map of New York State, at least within some area? Or if we need to fill up a bigger area, can’t we get a long enough strand to go filling it up? And then use whatever the number of the corresponding point on the strand is to describe the point in our area?
For that matter, if we wrap the yarn up on itself enough, obviously it fills space. That’s how balls of yarn work. That yarn ball fills up some space. Every point inside the bounds of the yarn ball … can’t we say the coordinate for a point of space inside the yarn ball is the coordinate for whatever bit of yarn occupies that space? And that we therefore need just the one number for this two- or three-dimensional region of space?
There should be no end of objections here. For one, yes, the yarn fills space, because the yarn isn’t a line. It’s a tube, and that a three-dimensional space can get filled up by a three-dimensional object shouldn’t be any kind of revelation. We have warnings going back several thousand years about using real-world examples as more than a guide to what things really are, particularly the reality of such difficult and impossible things as lines.
Even the physical example isn’t convincing. After all, there’s air between the threads, however densely packed it is. We can squeeze the air out, and can use a finer thread, and yet while we can still fill up the same size of yarn ball if we get enough thread, there’s still space between the strands.
We can imagine making an impossibly fine thread, at least after tripping over the question of whether we can imagine doing something we say up front is impossible. Can we imagine rolling it up so there’s no space between threads, though? Fine enough that the distance between any two threads is too small to measure, so there’s no missing space, and yet without the thread overlapping itself?
Well, there’s stuff called “space-filling curves”, which gives the answer away. Many of them can be built by taking a simple pattern and then iterating, turning what were line segments on the current pattern into small replicas of the simple pattern, and repeating this. It makes for great screen savers to watch, and probably has applications in the floor tiling department. And as promised we can fill as much space as we like using one continuous line, or thread, curved on itself incredibly many times.
It’s a fun stunt, but for the kind of things we like in coordinates it’s not very good. It’s natural to want points that are close to each other to have coordinates that are near each other, and that just doesn’t happen with this space-filling curve scheme. And we can look at a point and not have the faintest idea about what its coordinates might be. Coordinates have to do more than just exist to be useful.