Meditation on Some Fixed-Point Theorems

by John Derbyshire, 1945-

• **Background**

My 2006 book *Unknown Quantity* discusses
Brouwer's Fixed-Point Theorem, a very famous
result in mathematics. I comment
on one of the consequences of the theorem:

Place a sheet of paper on the desk and draw around its outline with a marker. Now scrunch up the sheet, without tearing it, and put the scrunched-up paper inside the outline you marked. It is absolutely certain that one (at least) point of the scrunched-up paper is vertically above the point of the desk that point rested on when the paper was flat and you were drawing the outline.^{151}

Endnote 151 reads as follows.

A related theorem, due to topologist Heinz Hopf (1894-1971), and often confused with Brouwer's FPT, assures us that at some point on the Earth's surface at this moment, there is absolutely, though instantaneously, no wind. Or equivalently, imagine a sphere covered with short hair, which you are trying to brush all in one direction. You will fail. No matter how you try, there will always be one (at least) "whorl point" where the hair won't lie down. This has led to the theorem being referred to rather irreverently by generations of math undergraduates as "the cat's anus theorem." (I have bowdlerized slightly.) Thus considered, the theorem states: Every cat must have an anus.

For some reason this theorem inspired me to verse.

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• **Play the reading**

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• **Text of the poem**

Upon the Earth's blue globe, at any date,

At any hour of day, there is a place

Where peace holds sway; the atmospheric state

Of perfect stillness dwells there for a pace.

And look! this crumpled paper, lightly tossed

Into a box it once served as a floor

Will yield a point which, though we thought it lost,

Lies straight above the place it lay before!

The one who grooms a cat may weep and pout

(as will the cat, with melancholy wail);

But, try his best, he'll never even out

That one accurséd spot beneath the tail.

Thus all things show what math can prove for sure:

No chaos is complete, no order pure.