Tuesday, December 27, 2011

43: A Theorem Made Real

    Did you know that at any given moment, on any given great circle
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
    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. 

  • Continuous Functions at Wikipedia
  • Intermediate Value Theorem at Wikipedia
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