Brouwer fixed-point theorem: Difference between revisions

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In [[mathematics]], the '''Brouwer fixed point theorem''' states that every [[continuous function]] from the closed unit [[Ball (mathematics)|ball]] ''D''^ ''n'' to itself has a [[fixed point (mathematics)|fixed point]]. In this theorem, ''n'' is any positive [[integer]], and the closed unit ball is the set of all points in [[Euclidean space|Euclidean ''n''-space]] '''R'''^''n'' which are at distance at most 1 from the origin.

The theorem has several "real world" illustrations. One common informal version is "you can't comb a hairy ball smooth"; another one works as follows: take 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 any way 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 applied 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.