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       #Post#: 115--------------------------------------------------
       WHY AN INTERNATIONAL DATA LINE
       By: eba95 Date: July 30, 2010, 6:47 am
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       Why an International Date
       Line?
       Can you find a continuous one-to-
       one function from the circle to the
       real line?
       Put it another way: is it possible, for
       every longitude on the earth, to
       assign a time so that each longitude
       has a different time, but the times at
       nearby longitudes are close?
       In fact, no. Any attempt to construct
       such a function will inevitably fail.
       This explains why the world has an
       "international date line": assigning
       time to geographical location is a
       function from a circle (longitude) to
       an interval (time).
       By convention, we normally assign
       times in discrete chunks (time
       zones), but the idea is the same. Our
       current method says that if it is 10
       AM on Wednesday in Claremont,
       then it is 11 AM in Denver, and 12
       noon in Minneapolis. Continuing
       around the globe in this manner,
       one finds that to keep nearby points
       having nearby times, one would
       have to assign Claremont a different
       time as well--- 10AM on THURSDAY!
       The only other alternative is to give
       up the continuity in favor of one-to-
       oneness, and put the discontinuity
       along a longitude of the earth that
       would affect the fewest people, i.e.,
       somewhere in the Pacific, and call
       the discontinuity a "date line".
       Presentation Suggestions:
       Do the presentation above (adjusted
       to your locale). Then read from the
       diary of Magellan's circumnavigation
       of the globe about returning to
       Europe (See Winfree, p.11):
       The 18 survivors of Magellan's
       expedition around the world were
       the first to present this dilemma for
       the bewilderment of all Europe. After
       three years westward sailing, they
       first made contact with European
       civilization again on Wednesday 9
       July, 1522 by ship's log. But in Europe
       it was already Thursday! Pigafetta
       writes (translated in The First
       Voyage Around the World, Hakluyt
       Society, vol. 52, 1874, p. 161):
       "In order to see whether we had
       kept an exact account of the days,
       we charged those who went ashore
       to ask what day of the week it was,
       and they were told by the
       Portuguese inhabitants of the island
       that it was Thursday, which was a
       great cause of wondering to us,
       since with us it was only
       Wednesday. We could not persuade
       ourselves that we were mistaken;
       and I was more surprised than the
       others, since having always been in
       good health, I had every day,
       without intermission, written down
       the day that was current."
       The Math Behind the Fact:
       The fact that there does not exist any
       continuous one-to-one function from
       the circle onto the interval follows
       from the Borsuk-Ulam Theorem in
       dimension 1. Topologists often study
       1-1 and onto functions which are
       continuous in both directions; such
       functions are called
       homeomorphisms and yield an
       equivalence relation for objects in
       topology.
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