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Borsuk-Ulam and West Wing

February 7th, 2006

In West Wing 4×20, 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 from the sphere to itself (meaning that f(-x)=-f(x) ) has odd degree. I won’t enter into what this means more precisely, but it has a few cool corollaries:

Any family of closed sets covering has a member containing an antipodal pair.
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 can be used to construct a function from the sphere to the plane by sending a point on the earth to the point (t(P),t(P)) . This function, in turn, has an antipodal fixpoint, so there are some points and such that (t(P),t(P))=(t(-P),t(-P)) . But this also means t(P)=t(-P) , 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 (t(P),t(P)) .

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 S^1 , and the temperature map takes S^1 to $\mathbb R$ , 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.

8 People had this to say...

Nick Said...

You don’t need topology to prove this. All that is necessary is that the temperature is a continuous function, and the result follows from the Intermediate Value Theorem. Generally, for any continuous periodic function f:R->R with period K, we define g(x) = f(x) – f(x – K/2).

Case 1: Suppose g(0) 0, and so g must equal zero in between, i.e. f must have equal value at two points exactly half a period apart.

Case 2: g(0) > 0. Same as above.

Case 3: g(0) = 0. Leads directly to our result.

For the Earth temperature example, f can be the temperature parameterized by distance (or whatever) on any closed path of your choice, not just great circle routes.

February 14th, 2006 at 15:13

Michi Said...

Well, yeah, I can see that argument. But it’s not as much fun as invoking the swiss army knife of Borsuk-Ulam!

February 14th, 2006 at 19:41

Nick Said...

I don’t even know Borsuk-Ulam, so the Intermediate Value Theorem was the best I could do. I often find simplicity appealing though.

February 16th, 2006 at 13:59

Scott Simmons Said...

One of my old professors proved, as his dissertation at CalTech, that there is some pair of antipodal points on the sphere that not only have the same temperature, but that can be connected by a continuous path on the surface along which every point is at that same temperature.

One of his professors at CalTech was the legendary physicist Richard Feynman. Just after our hero had finally finished this proof, he happened to run into Feynman on campus. “Sonneborn,”, he cried, “haven’t seen you in quite a while! What have you been up to?”

“Well, Professor, I’ve been working hard on my dissertation–I just finally worked out the last step in my proof!” Sonneborn was obviously pretty excited, he’d been working on this for quite some time. “You see, if you have a continuous function on a sphere … ” etc. etc. “The proof is really interesting …”

“Hold on!” commanded Feynman. Then he started muttering to himself, waving his hands around and making contorted faces. After a moment, he straightened up … and rattled off the whole proof, start to finish!

Sonneborn was in shock. Years of work on his dissertation to prove this, and Feynman worked it out in seconds! He wandered back to his office in a kind of daze, pondering the futility of thinking he might be able to be some sort of mathematician. His advisor found him there, staring blankly into space.

“Sonneborn! What’s wrong?” he asked. And the hapless grad student shakily told him the whole story. To his shock and disgust, his advisor broke down in hysterical laughter at the end. When he finally got his lungs back under control, he was able to explain what was so funny … “I just told him all about that at lunch!”

True story. According to the victim, anyway–I got it first-hand circa 1987 …

February 13th, 2007 at 1:02

Michi Said...

Scott: That’s an absolutely wonderful story. I love it!

And I have to steal that prank for later in my career…

February 13th, 2007 at 10:03

RaymonWazerri Said...

Hey,
I love what you’e doing!
Don’t ever change and best of luck.

Raymon W.

April 21st, 2007 at 0:32

J. Siehler Said...

You can use elementary stuff (IVT) to show there are antipodal points with the same temperature, but Borsuk-Ulam gives you more than that: you can consider a smooth function from the 2-sphere to R^2 (not just a real-valued function) and still be guaranteed a pair of antipodal points where the function takes the same value. For example, you can have same temperature AND wind speed at the same time. You will not necessarily find such a pair on every great circle; there may even be only one such pair.

January 4th, 2008 at 15:29

Michi Said...

J. Siehler: Yup, that’s right. And you might have noticed that this point has been made, repeatedly, in the preceding discussion.

January 4th, 2008 at 22:45

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about: Michi is a recent PhD working in homological algebra and applied algebraic topology. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
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