Why does every direction at the north pole point south?

Question

Why does every direction at the north pole point south?

Why doesn't this happen at any other point on (face of the) earth? Is this due to convention used by humans or is there a geometrical explanation to it?

The way I see it, at north pole the surface is flat like anywhere else, albeit somewhere in the arctic ocean and not land. If this is right, then why can't there be other directions like anywhere else?

I am not able to figure this out. Please explain the geometry behind it if it is applicable.

Answer 1

Assuming the Earth is a perfect sphere and the magnetic north pole is at the North Pole...

At any point on the Earth, any direction will "point towards the antipode", the point that is directly opposite on the sphere (the other point of intersection with the sphere of the line that goes through the original point and the center of the sphere).

It just so happens that the antipode of the North Pole is the South Pole, which means all directions are going "south" (i.e., "in the direction of the South Pole"). If you stand at the equator in the prime meridian, you will find that all directions "point to" the international date line equatorial point.

Answer 2

With the standard system of coordinates, latitude and longitude, if you are not at the north or south pole you can give your position on the earth uniquely. Notice that the longitude of the north and south poles are undefined. (This is no mistake, it is not possible to find a nicely behaved system of coordinates for the sphere with one chart. You need at least two.)

Anywhere except the poles you can give your direction of travel as an angle referenced to your line of longitude. When you say, "I am traveling northwest," this is what you mean. You have given the angle, it is $45^\circ$. (This is often written as N $45^\circ$ W.) If you know your latitude, longitude, and this angle, your journey is completely determined.

If you are at one of the poles, however, this method breaks down since your longitude is undefined. You must reference some other axis. It is natural to give your direction of travel by specifying which meridian you will take. You might say, "I am walking south along the Prime Meridian, and so expect to go through Greenwich on the way to the south pole."