If one stirs to dissolve a lump of sugar, it appears there is always a point without motion. He drew the conclusion that at any moment, there is a point on the surface that is not moving. The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit. The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.
Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet." Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. wikipedia
The most important fixed point theorem is Brouwer's (deals with functions); the extention of this theorem to correspondences is given by Kakutani's fixed point theorem.
Real world examples: (1) Take two equal size sheets of paper, one lying directly above the other. If you crumple the top sheet, and place it on top of the other sheet, then Brouwer's fixed point theorem says that there must be at least one point on the top sheet that is directly above the corresponding point on the bottom sheet. (2) Take a map of the city in which you live. Now lay the map down on the floor. There exists at least one point on the map which tells the location of the corresponding point below it on the floor.
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