http://blogs.reucon.com/ckienle/ http://blogs.reucon.com/ckienle/2008/01/22/1201022700000.html In preparation for my next test I came across a very fascinating mathematical problem. <br><br> "Prove that there always two subtended points on the Earth's equator where you are able to measure the exact same temperature." (Adapted from the Book: Repetiorium der Analysis Teil 1 (en. Revision Course for Analysis Part 1) from Timmann). <br><br> I have tried to visualize the proposition. The two red dots represent two subtended points on the equator. Remember: We want to show that there are always two subtended points (the red ones) on the Earth's equator where you are able to measure the exact same temperature. <br><br> <img src="http://www.christian-kienle.de/equator.png"> <br><br> If you believe it or not: That can be proven by combining simple mathematical concepts. <br><br> Let the points of the equator be the interval <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?I\; =\; \left[ 0,\; 2\pi \right]">, <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?p\; \in \; \left[ 0,\; 2\pi \right]"> a point on the equator and <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?T\left( p \right)"> a function which returns the temperature at the given point. <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?T\left( p \right)"> is a continuous function, obviously. <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?f\left( p \right)\; :=\;T\left( p \right)\; -\; T\left( p+\pi \right)"> returns the difference between two subtended points on the equator. Like the temperature function this function is continuous too. Because of <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?T\left( 0 \right)\; =\; T\left( 2\pi \right)\; is\; f\left( \pi \right)\; =\; -f\left( 0 \right)">. If <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?f\left( 0 \right)\; =\; -f\left( \pi \right)\; =\; 0"> then the two subtended points on the equator must have the same temperature. If <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?f\left( 0 \right)\; =\; -f\left( \pi \right)\; \neq \; 0"> then <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?f\left( 0 \right)\; and\; f\left( \pi \right)"> have different signs and therefore because of the "intermediate value theorem" there exists a <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?p\; \in \; \left[ 0,\; \pi \right]\; with\; f\left( p \right)\; =\; 0"> and this means that <img src="http://blogs.reucon.com/cgi-bin/mimetex.cgi?T\left( p \right)\; =\; T\left( p\; +\; \pi \right)"> which was we wanted to show. <br><br> When I first came across this problem I couldn't believe what one can prove with math. I hope you enjoyed my first post. More will come.