I've got a question:

For the discussion of math. Duh.

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Internetmeme
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I've got a question:

Postby Internetmeme » Tue Jun 16, 2009 6:28 pm UTC

I've been drawing while I was bored, and I had an idea.

I have a stack of paper on my desk, and I crumpled up the top and was about to throw it away when an idea struck me:

Doesn't there have to be at least one part of the paper that above the corrosponding part of the paper(s) under it?
Think of it this way:
Crumple a plane and put it on a plane, and there is at least one corrosponding point between the two. I'm sure someone else has thought of this if I could think of it. What's the name of this theory/theorem/postulate?
Spoiler:

mike-l
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Re: I've got a question:

Postby mike-l » Tue Jun 16, 2009 6:31 pm UTC

Yes. Look up fixed point theorems, one of them applies here.
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Ended
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Re: I've got a question:

Postby Ended » Tue Jun 16, 2009 6:42 pm UTC

A similar result (using the contraction mapping theorem) is that if you put a standard map of the world on the floor, exactly one* point on the map will be touching the point it represents on the earth.

*this may break down if you're standing somewhere (e.g. north pole) which is at the edge of your map, or if the map has repeated regions. I think in these cases you will get at least one point.
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Harg
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Re: I've got a question:

Postby Harg » Tue Jun 16, 2009 7:15 pm UTC

Yes, the Banach fixed point theorem works for the map. For the OP's question and the crumpled paper, the Brouwer fixed point theorem applies.

Buttons
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Re: I've got a question:

Postby Buttons » Tue Jun 16, 2009 7:24 pm UTC

Ended wrote:*this may break down if you're standing somewhere (e.g. north pole) which is at the edge of your map, or if the map has repeated regions. I think in these cases you will get at least one point.

You can get zero in certain situations. For instance, take a Mercator projection map that has the Americas on the left, and lay it across the 180th meridian (or whatever its western boundary is) with the north side facing north.

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quintopia
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Re: I've got a question:

Postby quintopia » Tue Jun 16, 2009 9:07 pm UTC

You will be very cold, though, since it seems the only place you can do this is in Chukot (I'm assuming here that the deck of a boat doesn't count as a particular point on Earth).

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Re: I've got a question:

Postby skeptical scientist » Tue Jun 16, 2009 9:22 pm UTC

Ended wrote:if you put a standard map of the world on the floor, exactly one* point on the map will be touching the point it represents on the earth.
Just look for the little red dot labeled, "You are here."
Buttons wrote:You can get zero in certain situations. For instance, take a Mercator projection map that has the Americas on the left, and lay it across the 180th meridian (or whatever its western boundary is) with the north side facing north.
This is because a map of the world is not exactly a contraction - it mostly is, but (depending on the projection) it may stretch the poles into lines, or map adjacent points onto opposite edges of the map. It's true if you have a globe though, or if you place the map in some region where locally it's a contraction.
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z4lis
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Re: I've got a question:

Postby z4lis » Wed Jun 17, 2009 3:36 am UTC

I'm not sure how to respond to this. My interest in mathematics is fueled by theorems like these. :)
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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Re: I've got a question:

Postby Buttons » Wed Jun 17, 2009 2:03 pm UTC

skeptical scientist wrote:This is because a map of the world is not exactly a contraction.

The biggest problem, of course, is that it's not continuous.

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MartianInvader
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Re: I've got a question:

Postby MartianInvader » Wed Jun 17, 2009 2:55 pm UTC

If we want a map of the world to be continuous, how about a beach-ball globe? Deflate the beach ball, crumple/stretch it however you want, then press it flat - two opposite sides of the world will always be right on top of each other (Borsuk-Ulam Theorem).
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!


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