Ok, I think this is neat.
Take a map of the world and lay it on the floor.
1. How many points on the map are exactly above the point on the ground that they represent? Prove it.
2. What about if you fold the map up before laying it on the floor? Or crumple it into a ball?
(A point y on the map is 'exactly above' a point x on the surface of the Earth if Oxy is a straight line, where O is the centre of the Earth.)
(Assume that the Earth's surface is 2D, and that the Earth is a perfect sphere, although I'm not sure that this makes any difference).
Fun with maps!
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Fun with maps!
Generally I try to make myself do things I instinctively avoid, in case they are awesome.
dubsola
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2.
Project the map onto the earth's surface. Then you get two images, one right where you are standing, and one on the opposite side of the earth. Each side must give you exactly one, whether the map is crumpled or flat, as it represents a contraction, which has at exactly one fixed point as it is a contraction of a metric space X (a large region around the point on the map) into itself.
The only exception to this is if a region cannot be found on the map containing the point right where you are standing, or it's opposite point, e.g. if you're standing on the north pole, and the north pole is at the border of the map. Then you need a different proof, but I'm pretty sure you still have 2 fixed points.
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"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson

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Yup, you got it.
It's an application of the contraction mapping theorem. I didn't think about the north pole case you give, but I think it still works as long as every point on the earth is represented on the map.
Generally I try to make myself do things I instinctively avoid, in case they are awesome.
dubsola
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Ended wrote:Yup, you got it.It's an application of the contraction mapping theorem. I didn't think about the north pole case you give, but I think it still works as long as every point on the earth is represented on the map.
Well, it's no longer a contraction, since two points very close together  e.g. 1 mm south of the north pole in the direction of greenwitch and in the direction of australia  can be mapped to points on the map which are much more than 2 mm apart. It may still be true, but you need a better proof. However, this is very unusual, since most points on the earth will not be points where the map is discontinuous. If you don't want to worry about this, just use a globe instead of a map.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
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imMAW wrote:If you buy a Really Huge map and align it correctly, there will be an infinite amount of matching points.
I think when he said map, there was an assumption it wasn't 11 scale. Which may not be valid, since 11 scale maps do exist in fiction, but it's a pretty reasonable assumption for most maps.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
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Are you sure this is right?
What if our map is a sort of weird projection from where we're standing, designed to preserve scale along exactly one line within that region of the earth, but is less stretched out elsewhere?
If this case isn't allowed, then it's trivial to show that (if we ignore the opposite side of the earth) at most one point on the map can be directly above its counterpart. If two points were directly above their counterparts, they would have to be an equal distance apart on the map and in reality, and then the map would definitionally be 11 scale along the line between them.
The trick is showing that AT LEAST one point on the map must be over its point on this side of the globe. I'm having some difficulty with that. It's easy in the 1dimensional case, but gets weird when I start considering polar projections and so on.
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Response to jwwells:
I was assuming that, by a "map", we could assume that there was some constant c less than 1 so that any two points are at least c times closer on the map as they are on the Earth. This isn't in the statement of the problem, but you'd be hard pressed to find a map of the earth for which it is untrue of points that aren't at the edges of the map (which I specifically cited as a potential problem in my solution.)
If this is the case, then the Banach fixed point theorem (link goes to Wikipedia) applies.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson

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This is my favorite map.
Ended wrote:2. What about if you fold the map up before laying it on the floor? Or crumple it into a ball?
Can't we get at least two points if we fold the map with a crease sufficiently close to our location? Or will the reduced scale of the map always make any additional points "beyond the reach" of the doubledover layer.
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