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!<\/p>\n\n\n\n
More formally, it says that any continuous function from an n<\/em>-sphere to Rn<\/em><\/sup> 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.)<\/p>\n\n\n\n Presentation Suggestions:<\/strong> The Math Behind the Fact:<\/strong> On an unrelated note, the Borsuk-Ulam theorem implies the Brouwer fixed point theorem, and there’s an elementary proof! See the reference.<\/p>\n\n\n\n How to Cite this Page:<\/strong> References:<\/strong> Fun Fact suggested by:<\/strong> The Borsuk-Ulam theorem is another amazing theorem from topology. An informal version of the theorem says that at any given...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[4,11],"class_list":["post-405","page","type-page","status-publish","hentry","tag-medium","tag-topology"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/405","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=405"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/405\/revisions"}],"predecessor-version":[{"id":1785,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/405\/revisions\/1785"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=405"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=405"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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.<\/p>\n\n\n\n
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!<\/p>\n\n\n\n
Su, Francis E., et al. “Borsuk-Ulam Theorem.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
F.E. Su, “Borsuk-Ulam implies Brouwer: a direct construction,” Amer. Math. Monthly, Oct. 1997.<\/p>\n\n\n\n
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"