around the world, somewhere there are two opposite, or antipodal,

points that are at the exact same temperature? When I first heard

this, I was a bit surprised. Why should it have to be that way?

Temperatures can vary a lot, and how are we to know how hot it is

somewhere on the opposite side of the Earth? Is someone watching a

cosmic thermostat to keep us in line? The strangest thing about this

fact is that we can prove it purely through math, without any

reference to chemistry, astronomy, or meterology.

Well, that's a bit of an exaggeration-- to prove that at any given

moment two antipodal points are at the same temperature, we need one

physical assumption: temperature is a continuous function with

respect to location. Intuitively, that means you can draw a graph of

temperature vs position without lifting your pen from the paper. In

other words, if it's 70 degrees where I'm standing, then if you're

standing pretty close to me, the temperature you feel will be pretty

close to 70-- there won't be a sudden discontinuous jump to 200

degrees. While continuity might seem like a relatively simple

property, it brings us a whole bunch of useful mathematical results

that we can now apply to our function.

In particular, if we have a continuous function, it is subject to

the Intermediate Value Theorem, an important 19th-century result

descending from the work of Bohemian mathematician and philosopher

Bernard Bolzano. It shows that if you have a continuous function f,

f(a) is less than 0, and f(b) is greater than 0, then somewhere there

must be a point c between a and b where f(c) = 0. When stated in this

pure functional form it sounds pretty obvious, doesn't it? If you

start out below 0, then get above 0 without any sudden jumps, you must

have passed through 0. Yet when we apply it to real-world situations

where a function is couched in other terms, the results can be

surprising.

So now let's go back to our original question: is it really true

that on any great circle around the globe, there have to be two

opposite points with the same temperature? Let's choose an arbitrary

great circle and pair of opposite points; for example, Buenos Aires

and Shanghai. Chances are, those two place don't have the same

temperature-- today it's 57 degrees in Buenos Aires, and 63 degrees in

Shanghai. Let's draw a line through the center of the earth between

them, and slowly rotate it around the great circle. Now let's define

our continuous function f as the difference in temperature at the

opposite ends of the line, vs. the angle we have rotated. f(0) is 63

minus 57, or 6. f(180 degrees) will be 57 - 63, or negative 6, since

we will have turned the line totally around. This means that by the

Intermediate Value Theorem, there must be some angle a for which f(a)

is 0. But then this means that for whatever angle a is, when the

rotating line is at that angle, it touches two antipodal points with

the same temperature!

So we have a surprising real-life consequence of what seems to be

a simple and obvious theorem, that a continuous function that is

sometimes greater and sometimess less than zero must hit zero at some

point. Now we know it's really true-- now matter how we draw a great

circle around the world, somewhere there are two opposite, or antipodal,

points that are at the exact same temperature. Perhaps you can amaze

your friends with this one at your next social gathering.

And this has been your math mutation for today.

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