According to the website Planetmath, “Especially malicious rumor has it that if you want to go by train from xx to yy in France, the most efficient solution is to reduce the problem to going from xx to Paris and then from Paris to yy.”
It is indeed only a rumor, but it’s not really too far off. Yes, you can get from Marseille to Nice in the south without going to Paris first, but just try to get from Bordeaux to Brest, both in the west, without traveling east to Paris first.
The role of Paris as a hub in the French railway system (SNCF) has led mathematicians, perhaps overly eager to show that they know something about the real world, to give its name to a special way of measuring distance in the two-dimensional plane.
Here’s how it goes: the point (0,0), also called the origin, is special, and to get between any two other points on the plane, you can only travel on straight lines to the origin. That means that if points a and b are on the same ray coming from the origin, you can travel from one to the other along that ray, the same way you would normally. But if they’re on different rays, you have to go all the way to the origin and then back out again.
As artificial as it seems at first, the SNCF metric is actually fairly reminiscent of some real-world travel situations. I’ve seen it called the British Rail metric and the Post Office metric, but it could just as easily be the Frontier Airlines metric, the Chicago Transit Authority metric, the "all roads lead to Rome" metric, or probably the metric from any number of other major cities or countries. Very rarely is the crow-flies distance between two points actually available as a way to get between them. On small scales, like in a neighborhood, another metric—sometimes called the taxicab metric—dominates. Basically, you can only travel on roads, so in an area with a grid system, the total distance between two points is the distance between their north-south coordinates plus the distance between their east-west coordinates. But on larger scales, things get more like the SNCF metric. We get to a hub—maybe an airport, train station, or major freeway—before going along a major route to our next destination.
But despite its resemblance to places and experience you’re familiar with, the mathematical space gets a little weird. In the real world, if you were standing a few feet away from a point but not on the same ray emanating from the origin, you’d just walk over. You wouldn’t have to take the train to Paris first. But the mathematical version of the SNCF metric is not as forgiving. For example, two points that are both 1 unit away from the origin, say the points (1,0) and (0.996,0.087), have a distance of 2 in the SNCF metric because if you're standing at one, you're not allowed to sidle on over to the other. You have to walk all the way to the origin first. In fact, any two points that are 1 unit away from the origin are 2 units away from each other.
A related property of interest to mathematicians is what points are within a certain distance of any given point, or as mathematicians would say it, what open sets look like in the space. In the usual way of measuring distance, everything within, for example, 1/4 of a unit from one given point forms a little round ball around that point. But in the SNCF metric, the neighborhoods are just little line segments, at least if you’re far enough away from the origin.
In a weird, squishy, topological way, that fact makes the plane bigger. Like the long line, which I wrote about in February, the SNCF metric is not second countable. That’s a fancy way of saying there’s no way to build the set using a small-ish (but still infinite) number of sets. The exact definition gets pretty technical, and to be honest, even a dozen years after my first topology class, I always have to look up the details anyway, so I’m not going to get into it here, but the basic problem is that open sets aren’t fat enough, so it takes too many of them to cover the space. Open sets in the usual distance metric are like splatter-paint. It’s not too hard to cover a canvas with that. Open sets in the SNCF metric are like thin hairs. How long would it take to fill in a canvas using a brush with a single hair?
I chose to write about the SNCF metric this month because I'm living in Paris right now, so I'm in on the joke. But it's also appealing because it gets me thinking about the way mathematics both does and does not reflect real-world situations. As far as I know, the SNCF metric was not intended to shed any light on French transit, but it's just fun to think about the absurdity of forcing any real place into an overly rigid mathematical model. Or maybe I need to stay in France a little while longer if that's my idea of fun. There are plenty of places I can visit by train!
Read about more of my favorite spaces:
The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Infinite Earring
The Line with Two Origins
The House with Two Rooms
The Fano Plane
The Möbius Strip
The Long Line
The Wallis Sieve
Two Tori Glued along a Slit
The Empty Set
The Menger Sponge
The Connected Sum of Four Hopf Links
The Sierpinski Triangle
Lexicographic Ordering on the Unit Square