The BorsukUlam 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 nsphere to R^{n} 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 1dimensional 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 BorsukUlam 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. "BorsukUlam Theorem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
