Brouwer Fixed Point Theorem

Drag the endpoints of the line. As long as no part of the dragged line falls outside of the range of the original line, at least one point will always lie at its original horizontal position, as shown by the indicator. The gray value of that point will match the gray of the background exactly. Try as hard as you like to find a dragged position where no point is in its original horizontal position; you will not succeed. The Brouwer fixed point theorem guarantees that.

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The surprising Brouwer fixed point theorem implies that if you crumple a map and place it on a copy of itself, at least one point on the crumpled map will be exactly on top of the corresponding point on the uncrumpled copy, no matter how it is placed; If you stir a cup of coffee, at least one point in the coffee will always be in its original position. You can gain an intuitive feeling for why that is so in this one-dimensional Demonstration of the theorem.
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