Brouwer fixed-point theorem: Difference between revisions

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# Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its 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 immediately beneath it.

# Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.

# In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a delicious cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume. Ordering a cocktail [[shaken, not stirred]] defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid). In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state. {{Citation needed|date=September 2018}}

==Intuitive approach==