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-sphere 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.)
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.