An Overused Intermediacy

I had wanted to talk about the Intermediate Value Theorem, since it’s one of those little utility theorems that doesn’t draw a lot of attention by itself but does have some wonderful results that depend on it. My context was in explaining just what Chiaroscuro had done when he figured out the fifth root of 1/6000th by guessing at it. I mean, he figured he was guessing at it, but there’s good reasons why this guessing would pay off and why he’d get to an answer near enough the right one.

And I wanted to talk about one of my favorite results of the Intermediate Value Theorem, at least as I remembered it: that at any time of the day or night, there must be at minimum a pair of antipodal sites — locations directly opposite the center of the Earth from one another — which have exactly the same temperature. Or the same humidity. Or the same of any meteorological measurement. I had read this, I was sure, in Richard Courant and Herbert Robbins’s masterpiece of mathematics writing, What Is Mathematics? and went digging about to find it precisely stated, particularly since as I remembered it was possible to get any pair of measurements — say, temperature and humidity together — exactly equal at antipodal sites.

Sad for me: if I did get this example from Courant and Robbins, it’s in a section I can’t locate now. And searching online revealed one of those humbling little moments: this antipodal-equality property was never this interesting little discovery that had caught my mind. It’s caught many, many people’s minds, and I’ll never even rank among the first ten thousand of people Google turns up to write about it. A shame. If this double-equality version exists I haven’t found it, although I admit my heart went out of the searching.

At least there are still some similar problems (cribbing again from Courant and Robbins) that don’t appear to be quite as well-worn, and are at least as surprising. For one of them: imagine you draw two shapes, not connected to one another, on the same sheet of paper. They can be any size you like, they can be any shape you like, as irregular or as blobby as you like, as long as they don’t touch one another. Then, it’s always possible to draw a single line which simultaneously cuts both shapes into equal-area halves. It’s easy to prove that bisecting line has to exist, although there’s not much help given in finding it.

Here’s another version that’s at least as remarkable, from the same section: draw any kind of shape you like, as irregular or as smooth as you like. It’s always possible to draw a pair of perpendicular lines, the classic + type crossing of lines, so as to divide the blob into four sections of equal area. I offer no help in telling you where to place the lines, or how to orient them relative the edge of the paper or the longest dimension of the blob, apart from whatever help it is to know there is always a way to do it. Do enjoy.