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if I wanted to show someone where in England I am I could just point to a spot on this map and that point is lying exactly above the place that represents my kitchen table I could do the same if I were living in London or in Edinburgh as long as I am somewhere in England there is a point on this map that is in exactly the same location as the point in England that it represents mathematically this is called browers fixed point theorem and Brouwers theorem also says that if I stir this cup of tea no matter how much I stir it there will always be one bit of liquid in exactly the same spot as it was before I started stirring points that stay the same after some transformation occurred fixed points browers fixed point theorem says that every continuous map from a compact convex set to itself has a fixed point lets deform this square parts where the square have moved a lot a colored in red parts that have not moved at all are black as long as it remains within the original bounds no ma