Comments on: Borsuk-Ulam and West Wing

By: Michi

Michi — Fri, 04 Jan 2008 21:45:40 +0000

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

By: J. Siehler

J. Siehler — Fri, 04 Jan 2008 14:29:40 +0000

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.

By: RaymonWazerri

RaymonWazerri — Fri, 20 Apr 2007 23:32:06 +0000

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

Raymon W.

By: Michi

Michi — Tue, 13 Feb 2007 09:03:43 +0000

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

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

By: Scott Simmons

Scott Simmons — Tue, 13 Feb 2007 00:02:54 +0000

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 …

By: Nick

Nick — Thu, 16 Feb 2006 12:59:26 +0000

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

By: Michi

Michi — Tue, 14 Feb 2006 18:41:46 +0000

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

By: Nick

Nick — Tue, 14 Feb 2006 14:13:45 +0000

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.