The Brouwer Fixed Point Theorem

When I heard about this theorem as an undergrad (yes, I was a member of the Math Club) I was initially nonplussed.  There was no way this claim could be true.  I’m still not sure it could be true.  Mathematicians have proven it is true.  I (attempt to) follow the proof that it is true.  Sure enough, it’s true.  Skipping the math, here is the cool outcome of this theorem:

If I take a map of a place, crumple it up, and throw it on the ground, there will be at least one point on the map that is exactly on top of the very point it represents in the real world.

Got it? One more time:

Take a map. Crumple it any way you please. Throw.  One point on the map is now exactly on top of the point it represents.

Are you at work, in your office?

Take the floorplan. Drop it on the floor in your office.  There is now a point on the floorplan which corresponds to it’s exact location.

Are you at home? Making dinner?

Take a map of the US. Fold it. Throw it in with the spaghetti.  There is now a point on Rand McNally himself that is touching the very piece of spaghetti, at the exact location of that spaghetti, at that exact location on the map.  Yum.

If this does not impress you, think about it more.  If you think this theorem is obvious or trivial, you are either a genius, a topologist, or an idiot who doesn’t really get what is happening.  I think it’s pretty neat.

 Since my blog posts get no respect unless they contain a lolcat, here is a mildly relevant lolcat to support my claims.

 

mapquest.jpg
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~ by wcuk on February 14, 2008.

5 Responses to “The Brouwer Fixed Point Theorem”

  1. *LoLcat +1
    *A bit obvious +0
    *Tried this with a map of Spain, and just got a crumpled, spaghetti-soaked map. -$5.99 Oh, and the Pyrenees are covered with tomatoes, not snow.
    *Thought about mapping a given point in real space onto the now crumpled map and wondered if a recursive algorithm would unfold a mapping of the real world onto the map as each successive base case was executed. +/-?
    *Decided that I need to appreciate your observation as one appreciates a LoLcat and go to sleep. +2.5

  2. *charlie proposing a recursive algorithm. +INF

    flawless victory?

  3. I’m not sure I get how said algorithm would work, but I award you each +threeve points.

  4. I’m not worthy of adding to this discussion

  5. Take a map of the world, crumple it up, and throw it on the ground. If you are in the Netherlands there is a chance a point on the crumpled map is on top of the point where ancient Dutch cartographers brutally killed Brouwer for wasting maps.

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