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Kakutani's theorem

This post on Kakutani's theorem broke my brain for a bit:

Take two pieces of 8*11 paper and lay them on top of one another so that every point on the top paper corresponds with a point on the bottom paper. Now crumple the top piece of paper in anyway that you wish and place it back on top. B's theorem tells us that there must be a point which has not moved, i.e. which lies exactly above the same point that it did initially.

Comments (3)


Makes sense, though. If, to take an extreme example, I crumple the top paper into something with single-point density, then all of its constituent points exist in that space. As long as it must remain over some part of the bottom paper, then naturally one of the "points" in the top paper is going to still be above its original location, because all of the top points are now over one of the bottom points.

Maybe because I went to that example first to test the idea, it seemed less brain-breaking to me. It's actually a little more disconcerting to imagine crumpling the paper up so that it has a footprint of only half the size of the lower paper, then considering that it will still have at least one mapping point.

Nifty stuff.

kwc Author Profile Page:

The monk travelling up and down the hill makes sense to me, but the paper one for me escapes visualization. I'll have to think a bit more from your perspective.

Dick Petersen:

It seems to me a one compress the space of a sheet of paper A by crumpling it up to a space 1/2 the size and perform this function infinitely the crumpled space will become a point set where one of the points in that set is correlated over the same point in the space A.

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