Brouwer fixed-point theorem: Difference between revisions

Revision as of 22:59, 5 September 2001

Revision as of 14:01, 6 September 2001

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The theorem has several "real world" illustrations. Take for instance two equal size sheets of graph paper with coordinate systems on them, lay one flat on the table and crumple up (but don't rip) the other one and place it anyway you like on top of the first. Then there will be at least one point of the crumpled sheet that lies exactly on top of the corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the ''n'' = 2 case of Brouwer's theorem sinceapplied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet right beneath it is continuous.