Can A Ball Of Yarn Threaten Three-Dimensional Space?


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.

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Searching For Infinity On The New York Thruway


So with several examples I’ve managed to prove what nobody really questioned, that it’s possible to imagine a complicated curve like the route of the New York Thruway and assign to all the points on it, or at least to the center line of the road, a unique number that no other point on the road has. And, more, it’s possible to assign these unique numbers in many different ways, from any lower bound we like to any upper bound we like. It’s a nice system, particularly if we’re short on numbers to tell us when we approach Loudonville.

But I’m feeling ambitious right now and want to see how ridiculously huge, positive or negative, a number I can assign to some point on the road. Since we’d measured distances from a reference point by miles before and got a range of about 500, or by millimeters and got a range of about 800,000,000, obviously we could get to any number, however big or small, just by measuring distance using the appropriate unit: lay megaparsecs or angstroms down on the Thruway, or even use some awkward or contrived units. I want to shoot for infinitely big numbers. I’ll start by dividing the road in two.

After all, there are two halves to the Thruway, a northern and a southern end, both arranged like upside-down u’s across the state. Instead of thinking of the center line of the whole Thruway, then, think of the center lines of the northern road and of the southern. They’re both about the same 496-mile length, but, it’d be remarkable if they were exactly the same length. Let’s suppose the northern belt is 497 miles, and the southern 495. Pretty naturally the northern belt we can give numbers from 0 to 497, based on how far they are from the south-eastern end of the road; similarly, the southern belt gets numbers from 0 to 495, from the same reference point.

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Searching For 800,000,000 On The New York Thruway


So we’ve introduced, with maybe more words than strictly necessary, the idea that we can set up a match between the numbers from 0 to 496 and particular locations on the New York Thruway. There are a number of practical quibbles that can be brought against this scheme. For example: could we say for certain that the “outer” edge of this road, which has roughly the shape of an upside-down u, isn’t loger than the “inner” edge? We may need more numbers for the one side than the other. And the mile markers, which seemed like an acceptable scheme for noting where one was, are almost certainly only approximately located.

But these aren’t very important. We can imagine the existence of the “ideal” Thruway, some line which runs along the median of the whole extent of the highway, so there’s no difference in length running either direction, and we can imagine measuring it to arbitrarily great precision. The actual road approximates that idealized road. And this gives what I had really wanted, a kind of number line. All the numbers from zero to 496 (or so) match a point on this ideal Thruway line, and all the points on this Thruway match some number between zero and 496. That the line wriggles all over the place and changes direction over and over, well, do we really insist that a line has to be straight?

Well, we can at least imagine taking this “ideal” Thruway, lifting it off the globe and straightening it out, if we really want to. Here we invoke a host of assumptions even past the idea that we can move this curvy idealized road around. We assume that we can straighten it out without changing its length, for example. This isn’t too unreasonable if we imagine this curve as being something like a tangled bit of string and that we straighten it out without putting any particular tension on it; but if we imagined the idealized road as being a rubber band, held taut at the New York City and Ripley, New York, ends and pinned in place at the major turns we notice that isn’t actually guaranteed. Let’s assume we can do this straightening-out without distorting the lengths, though.

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Searching For e On The New York Thruway


To return to my introduction of e using the most roundabout method possible I’d like to imagine the problem of telling someone just where it is you’ve been stranded in a broken car on the New York Thruway. Actually, I’d rather imagine the problem of being stranded in a broken car on the New Jersey Turnpike, as it’s much closer to my home, but the Turnpike has a complexity I don’t want distracting this chat, so I place the action one state north. Either road will do.

There’s too much toll road to just tell someone to find you there, and the majority of their lengths are away from any distinctive scenery, like an airport or a rest area, which would pin a location down. A gradual turn with trees on both sides is hardly distinctive. What’s needed is some fixed reference point. Fortunately, the Thruway Authority has been generous and provided more than sixty of them. These are the toll plazas: if we report that we are somewhere between exits 23 and 24, we have narrowed down our location to a six-mile stretch, which over a 496-mile road is not doing badly. We can imagine having our contact search that.

But the toll both standard has many inconveniences. The biggest is that exits are not uniformly spaced. At the New York City end of the Thruway, before tolls start, exits can be under a mile apart; upstate, where major centers of population become sparse, they can spread out to nearly twenty miles apart. As we wait for rescue those twenty miles seem to get longer.

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