Lines That Cross Infinitely Many Times
MathematicsHub is a new mathematics blog. Among its first entries was an effort to answer why two straight lines in space can intersect in at most one point. That is, if two lines intersect at more than one point, they have to intersect at all of them. And that’s indistinguishable from being the same line.
If we suppose we’re talking about ordinary old space and straight lines and intersections, then that’s true and there’s not much arguing it. But I got to wondering about non-ordinary spaces and lines and such. Could we work out something that looks like a pair of lines, and that intersect in more than one place and still aren’t the same line?
There’s an obvious answer. That comes in spherical geometry, the way shapes on the surface of a ball work. In this space, “lines” are instead “great circles”. Those the equator, lines of longitude, the paths that airplanes would travel if they didn’t have to deal with winds or restricted airspaces or radar paths. And two distinct great circles will intersect in two points, which you can convince yourself of by looking at the south and north poles of a globe.
Can we come up with something where lines intersect at three points? Or four? … Possibly. If we started with something ellipsoidal, perhaps. Or if we started with a sphere and pinched off a corner and twisted it around maybe we could make most “great circle” routes intersect several times. At least, I can imagine this. I admit I don’t have the background in non-Euclidean geometries to make a compelling case for it. It feels right to my instincts, is all, and I leave it as a homework problem for someone who wants homework problems.
It struck me I could think of something that’s very line-like and that offers infinitely many intersections, though. It’s again on a sphere, like the surface of the Earth. Take a path that has a constant compass heading. That is, something that’s always (say) 30 degrees counterclockwise of the south-north line. Following this path, always that same angle with respect to the local south-north line, creates a path called a “loxodrome”. It will look very much like a straight line, and on a Mercator-projection map of the world will be a straight line.
However, if you look at the sphere, this loxodrome traces out a path that spirals in towards the south and the north poles. Properly, it never reaches the south or north pole (unless your loxodrome was pointing directly south or directly north all the time). It just keeps looping around, infinitely many times.
And there’s my idea. Suppose we accept a loxodrome as being enough like a line. I admit you might not be willing to go along with that. If you’re not willing to make this supposition then you won’t accept my conclusion and that’s that. We’ll just have to disagree. (And I’d grant that in most cases I wouldn’t call a loxodrome a line, because it doesn’t behave enough like a line for most purposes.)
But if you will let me call loxodromes a kind of line, then I can give you this happy conclusion. If you start two loxodromes from the same point and going in different angles, then, they’re going to intersect. And not just once, nor twice, but many times as you go closer to the south or the north pole. And despite having infinitely many intersections, they’re not the same loxodromes. They point in different directions and touch many different points.
And isn’t that remarkable?
You may not be convinced. It depends whether you’ll accept a loxodrome as being enough like a line. But I like the idea of lines that intersect infinitely many times yet aren’t the same line. And let me know if you don’t buy this, and why; or if you have a better idea, please.