Borsuk-Ulam and West Wing

In West Wing 4x20, CJ states that there are two antipodal points with identical temperature on the earth, as an argument why it should be possible to imagine that an egg could stand on its end at the spring equinox. This particular plotline also has her most emphatically claiming that this should not be possible at the autumn equinox. Why this particular physics is complete idiocy will be left as an exercise to the interested reader, and instead I will focus on the other claim.

This is, in fact, true. It's a corollary to one of the prettiest theorem conglomerates I have ever seen: the Borsuk-Ulam theorem(s). Alas, I haven't got my sources on it here at the moment, so I won't give you the deep indepth survey I want to give; but I do want to give a bit of overview as to why the claim CJ supports her insane theory with is actually true.

Temperature can be seen as a function from location to the thermometer. For each place on the earth, there is a temperature rating. Furthermore, this function is continuous, since there are no discontinuities - no sudden jumps in temperature between close points.

This sudden jump thing merits closer explanation. It says that by measuring closer and closer together, you can get the difference in temperature as small as you want to. It doesn't prevent steep temperature shifts, it only prevents insane. This assumption, as such, is a valid one since temperature differences tend to flatten themselves out - if you place a hot and a cold bit close to each other, the cold one heats and the hot chills.

So, we know that it's continuous. This, it turns out, is an incredibly powerful to know. It brings in the entire toolbox of topology. Which, in turn, brings us closer to the topic of the post - the Borsuk-Ulam theorem.

The theorem has a million different equivalent statements. Borsuk himself proved that for any antipodal function [tex]f[/tex] from the sphere to itself (meaning that [tex]f(-x)=-f(x)[/tex]) has odd degree. I won't enter into what this means more precisely, but it has a few cool corollaries:

  1. Any family of [tex]n+1[/tex] closed sets covering [tex]S^n[/tex] has a member containing an antipodal pair.
  2. Any map from the sphere to the plane must send some antipodal pair to the same point

1 has a few fun interpretations. I will feed you with those when I get hold of my material again, or actually, you know, think of them.

2, on the other hand, is what we need to prove CJs statement. Our temperature function [tex]t[/tex] can be used to construct a function from the sphere to the plane by sending a point [tex]P[/tex] on the earth to the point [tex](t(P),t(P))[/tex]. This function, in turn, has an antipodal fixpoint, so there are some points [tex]P[/tex] and [tex]-P[/tex] such that [tex](t(P),t(P))=(t(-P),t(-P))[/tex]. But this also means [tex]t(P)=t(-P)[/tex], so there are some pair of antipodal points on the earth with the same temperature.

Thanks to nerdy2@#math:ircnet for reminding me of the details of Borsuk-Ulam, and giving the construction of [tex](t(P),t(P))[/tex].

Edit: It was pointed out to me later that all this is really unnecessary. In fact, we can use Borsuk-Ulam straight off to show that on each great circle on earth, there is such an antipodal pair. Indeed, a great circle is [tex]S^1[/tex], and the temperature map takes [tex]S^1[/tex] to [tex]\mathbb R[/tex], and thus qualifies for the setup for 2 above, since the dimensions of "sphere" and "plane" don't really matter. Thus, on each great circle there is an antipodal pair with equal temperature. And since there are many, many great circle's on earth, there are also many, many such points.