Why is North North?

Why is north North?

My cousin recently asked an interesting question:

Why can you travel east indefinitely but you can only travel north until you reach the north pole?

 As with all good questions, there are actually a bunch of good questions here all packed in together. As is my wont, I'm going to go way too deep, and hopefully come out the other side of this question with some answers. Chalk this up under the dangers of asking Luke a question.

What is North?

First, what is North, and why? How is that particular direction defined, and why is it useful?

The answer begins with a cool fact about rotations. Any constant rotation in three dimensions leaves an axis, or a line through space, fixed in place while everything else moves around it. Try rotating a few objects to see what this means. It's a law of geometry that this is so, and a law of physics that spinning objects prefer this kind of motion.

When objects experience constant rotation there is always a line through the object that doesn't move.

The earth spins (surprise!) and so there is an axis that is left fixed in space that passes through the center of the earth and picks out two special points where it intersects the surface. These two points are called poles, and we can use them to define directions.

Here's how the system works (it may be obvious, but I wouldn't be writing this if it weren't for pedantry): First name the two poles North and South. This is arbitrary as long as you keep them straight, which you can do using the fact that the stars look different in the northern hemisphere than in the south. Now at any arbitrary point on the earth, you can draw lines that directly connect that point and the two poles. Draw arrows on the lines that point towards the North pole. Now when you are walking towards the North pole on the most direct line you are walking North, and likewise for South. This is the definition.

How North is defined: at every point on earth, draw the most direct line to the north pole. The direction of the resulting lines is called North.

You might ask "how can I tell if I'm moving towards the pole if it is far away from me?" Good imaginary question! The answer goes back to the rotation of the earth. That fixed axis that has been useful also ensures that we can orient ourselves to a pole no matter where we are on the earth as long as we can see the night sky. If you look at the stars over a long period they appear to move because of the rotation of the earth. And the stars nearest the axis move the least. If you face the spot in the sky where the stars don't move at all then you are facing one of the poles. Useful!

A time lapse showing the rotation pole in the night sky. Courtesy http://www.eso.org/public/images/potw1534a/ via Creative Commons.

Even better, you can measure how far away you are from the pole with a pretty simple measurement.
Since the earth is a sphere you can measure locations with angles instead of miles or whatever. In particular, you might want to know the angle between your location and the pole. The diagram below hows how you can measure this angle simply by measuring the angle between the horizon and the pole spot in the sky.

You can find your latitude (how far you are from the North pole) by finding a pole star and measuring the angle it makes from the horizon.

All together, this is how North is defined, and why it is defined that way. (And I haven't even mentioned how the earth's magnetic field also runs North/South.)

 Singularity (is a cool word)

If you've been practicing your pedantry you may have already noticed slight problem with the definition of North. There are two points on the earth where the definition fails. Since we defined North as the direction towards the North pole, North doesn't mean anything when you are standing at the pole! Also, since South is the shortest direction towards the South pole, every direction is South when you're at the North pole, so it's not defined either. Both poles have this problem.

A singularity is where a system of directions stops working. North and South both have singularities.

A point where a direction scheme fails like this is called a coordinate singularity. But note these singularities are not problems with the earth! You can cross the poles all you want. The problem is with the scheme we've defined for describing directions. You can go any direction from the poles or walk over them just fine, you'll just have problems describing what you're doing in the standard North/South scheme.

We'll come back to singularities later. But at least now you feel cool for reading something that legitimately uses the word singularity.

East!

Now that we've defined North, East is pretty easy. Face North, then point to the right at a 90 degree angle to the North/South line. That's it.

The direction East is defined at a point to be at a right angle relative to North at that point.

If, for some reason, you can't distinguish right and left, wait until night. When the sun rises in the morning, East is the direction that is perpendicular to North and points towards the sun.

Of course, this definition also has singularities. Because it is defined relative to the direction North, it fails anywhere that definition fails. So you can't go North at the North pole, and you can't go East either.

Are we done yet?

Now the original question, at long last! Both North and East have singularities where the directions become undefined. The difference between North and East is that North by definition leads you towards the singularity, while East by definition keeps you the same distance away. You can't go North forever because you will always eventually reach a point where North is meaningless. You can go East forever because you will never reach the singularity.

Can we do better?

 One might ask if there is a better system, one that doesn't have confusing singularities so that all directions are defined everywhere. The answer is no, not really, because of something called the hairy ball theorem (seriously). But I'll leave that for another time.

Heh heh, hairy ball.