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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.

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There is nothing special geometrically (in terms of a sphere) about a North Pole. But South is defined effectively as "directly away from North (the North Pole)". –  Ronald May 6 '12 at 20:26
    
A somewhat related question –  t.b. May 6 '12 at 23:22
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3 Answers

up vote 7 down vote accepted

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.

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So if I happen to be standing in Trafalgar Square, every direction must point towards the antipode somewhere on the other side, hence only one direction, right? Why then do we identify more than one direction? –  Stu1 May 6 '12 at 20:39
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@Stu1: By definition, "north direction" means that if you advance along the corresponding great circle, you will get to the (magnetic) North pole before you get to the south pole. "South direction" means you will get to the south pole before you get to the North pole. "East" means you advance perpendicular to the north/south direction with "north direction" on your left, and "west" means you advance perpendicular to the north/south direction with "north direction" on your right. Since all directions are relative to "the direction along which you will get to the North pole" (cont) –  Arturo Magidin May 6 '12 at 20:45
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@Stu1: (cont) it is not surprising that they stop making sense when there is no such direction. But they make sense everywhere else. If you are in the North Pole, there is no direction along which you will get to the North Pole before you get to the South Pole. You will always get to the South Pole first. So there is not "going North". You cannot go towards your house if you are already in your house. –  Arturo Magidin May 6 '12 at 20:47
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"magnetic" has nothing to do with this, unless you're trying to use a compass to tell your directions (which is a bad idea in the Arctic and Antarctic). At the north pole, your compass will point south, which isn't saying much. Near the magnetic north pole, your compass is completely useless. –  Robert Israel May 6 '12 at 22:31
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A quibble: the International Date Line is not the same as the 180° meridian; the IDL wiggles all over the place. –  MJD May 6 '12 at 22:37
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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."

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Does your answer suggest that when you set out from a point on (the surface of) a sphere in a given direction you are always going toward this point's antipode? If so, would you have to steer (change direction) in order to travel in a path other than a great circle? What about travel along, for example, a line of constant latitude other than the equator? Is it possible to pick a single direction from a point on a sphere that does not result in a path which is a great circle?

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