If you haven’t seen the Aardman Animation movie Arthur Christmas, first, shame on you as it’s quite fun. But also you may wish to think carefully before reading this entry, and a few I project to follow, as it takes one plot point from the film which I think has some interesting mathematical implications, reaching ultimately to the fate of the universe, if I can get a good running start. But I can’t address the question without spoiling a suspense hook, so please do consider that. And watch the film; it’s a grand one about the Santa family.
The premise — without spoiling more than the commercials did — starts with Arthur, son of the current Santa, and Grand-Santa, father of the current fellow, and a linguistic construct which perfectly fills a niche I hadn’t realized was previously vacant, going off on their own to deliver a gift accidentally not delivered to one kid. To do this they take the old sleigh, as pulled by the reindeer, and they’re off over the waters when something happens and there I cut for spoilers.
What happens is: Arthur and Grand-Santa fall out of the sleigh, and watch it go flying off past the horizon, leaving all potentially doomed. Grand-Santa comes to explain that, without anyone guiding them, the relentless and somewhat stupid reindeer will just keep flying in a straight line. So … what should happen?
I think the first thing we can rule out is the idea that they are literally going to continue on a straight line forever and ever, as this would be the path of a tangent to the sphere of the Earth. The tangent — the word comes from “to touch” — touches the curve of the earth (well, a little above, as they were flying at the moment) in one point, at least locally, and continues on straight which means it’d be going off into space, roughly speaking along the path that the center of a headlight mounted on front would follow.
(We start getting into really complicated territory if we point out the curvature of space, and then start wondering how we know whether something is straight, and I’m ignoring all that as too much a complication for my purposes. I’m also ignoring the fact that the Earth isn’t a perfect sphere, but is lumpy enough to be measured even by 18th century geodesy techniques. They wouldn’t make a substantial change besides requiring that I use more words, which you’d rather I didn’t. Until I’m paid by the word I’d rather I didn’t too.)
So, if the reindeer followed a perfectly straight line, then from the perspective of someone sitting on the Earth’s surface, they’d keep rising up, going off into space. This is obviously absurd behavior to expect from a team of magic reindeer. Therefore we conclude that whatever Grand-Santa meant exactly by flying in a straight line, he didn’t mean they would fly the tangent line. We need to think of what he did mean instead.
The next obvious thing he might have meant: the reindeer might keep a constant compass heading. Constant compass headings — the navigational tool which made Mercator-projection maps such a hit — make it relatively easy to get where one wishes to go, simply by keeping the angle of one’s path, with respect to latitude or longitude lines, steady. Of course, reindeer have no reason to know of the existence of latitudes, past maybe a rough sense of how high the sun and moon should be, and none for the existence of longitudes unless they’ve got a natural instinct for astronomy or a good sense of what the time is back home (which, being at or near the North Pole, presents other problems).
But we can rationalize around that. Birds and butterflies do very well navigating, including over extended paths over the sea. We can give the same abilities to magic reindeer. It wouldn’t be unreasonable for flying reindeer to have a magnetic sense, for example, and I can’t declare it impossible they wouldn’t be able to sense the increases and decreases of magnetic force. This could provide them with a way to travel relative not quite the lines of latitude, but at least relative the curves of constant magnetic field strength. It’s a similar idea, and a similar covering of the globe, albeit centered on the magnetic poles rather than the rotational poles. Perhaps combining a sense of magnetic field lines and the sun’s position — in principle, a guide to latitude — would let them pick out latitude and longitude precisely. Such a plan was proposed by Edmund Halley, of Comet fame, as a solution to the problem of navigation, and while it was impractical for sailing ships that doesn’t mean nature might not do better.
So what happens if they follow a constant compass bearing? Here I’m sorry I haven’t got a sheet of Mercator-ruled lines to show off, since the result is a little surprising, but here’s a page which discusses them as well as some of the many issues involved in projecting a globe onto a sheet of paper (search for “plumb line” or “loxodrome”. I know the applet I want, if people still speak of applets, would be one letting you pick an arbitrary point and direction and showing where that line takes you, though). A constant compass bearing is a straight line on the Mercator projection, and unless you’re following a line of latitude or a line of longitude to start with, that straight line will just keep going on and on. This path, seen on the sphere, is a curve known as the “loxodrome”, and it spirals on in to the pole. In principle it spirals infinitely many times as one keeps getting closer and closer to the pole, but I would expect in practice that once the reindeer got near enough home they’d find their stable and settle back down.
If the reindeer were heading roughly North, then, they’d find their way eventually to the North Pole and one could assume that someone in the Santa Claus organization would figure out what was going on. If the reindeer were heading roughly South, they’d get to the South Pole and who knows what might go wrong from there. If they were heading due East or due West, they’d eventually circle around back to where they started, though. In the movie at that point, they were almost certainly heading northward (barring local course corrections), so I expected the sleigh to make its way to the North Pole.
There are more things which could happen, though.