Borsuk-Ulam Theorem

The Borsuk-Ulam theorem is another amazing theorem from topology. An informal version of the theorem says that at any given moment on the earth’s surface, there exist 2 antipodal points (on exactly opposite sides of the earth) with the same temperature and barometric pressure!

More formally, it says that any continuous function from an n- to Rn must send a pair of antipodal points to the same point. (So, in the above statement, we are assuming that temperature and barometric pressure are continuous functions.)

Presentation Suggestions:
Show your students the 1-dimensional version: on the equator, there must exist opposite points with the same temperature. Draw a few pictures of possible temperature distributions to convince them that it is true.

The Math Behind the Fact:
The one dimensional proof gives some idea why the theorem is true: if you compare opposite points A and B on the equator, suppose A starts out warmer than B. As you move A and B together around the equator, you will move A into B’s original position, and simultaneously B into A’s original position. But by that point A must be cooler than B. So somewhere in between (appealing to continuity) they must have been the same temperature!

On an unrelated note, the Borsuk-Ulam theorem implies the Brouwer fixed point theorem, and there’s an elementary proof! See the reference.

How to Cite this Page: 
Su, Francis E., et al. “Borsuk-Ulam Theorem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
F.E. Su, “Borsuk-Ulam implies Brouwer: a direct construction,” Amer. Math. Monthly, Oct. 1997.

Fun Fact suggested by:
Francis Su

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