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One of the great problems of classical geometry was to construct with straight-edge
and compasses only the trisection of an angle.
Bisecting an angle is fairly easy and is
usually taught right after bisecting a given line segment. It turns out it is impossible to
exactly
trisect an arbitrary angle given those tools. This is rather sad as it means one
cannot construct a nonagon (regular nine-sided polygon). That also rules out
regular
polygons with 18, 36, 72 ... sides as well as polygons with 27, 81, 243 ... sides.
What regular polygons can be constructed? The Greeks listed
triangles, squares,
pentagons, hexagons and pentadecagons (15 sides). Since one can always bisect a side
that meant 8, 16, 32 ... siders could be
constructed from squares; 12, 24, 48 siders from
hexagons; 10,20, 40 siders from pentagons and 30, 60, 120 siders from pentadecagons.
Notable by their
absence were polygons with 7, 11,13, 17, 19 ... sides. There were no
exact constructions known, but no one had proved it was impossible. For two thousand
years there were many ingenious attempts to trisect angles. To some extent, that quest
continues today, although usually serious geometers acknowledge that the
classical
construction is impossible before showing some new tactic that uses a supplementary
tool or improves the accuracy of a nearly
perfect construction. The problem is that
as powerful as a straightedge and compasses are, the best they can do is square roots,
and what is needed in
trisections is the ability to construct a cube root. One of the
better detailed treatments is the book The Trisection Problem by Robert Carl Yates
(1904-1963) , a Professor at the University of South Florida and a former supervisor of
instruction at the United States Military Academy at West Point. Speaking of which,
during much of the twentieth century, the Military Academy was a
leader on the
football field. After winning the Army-Harvard football game 46-0 on November 5,
1932 Academy Superintendent Connor and Harvard
President Lowell were at the home
of Mrs. William Putnam. Lowell asserted Harvard would win any academic contest. It
was decided that a
mathematical contest would take place at West Point May 19 and
20, 1933 with ten sophomores from each school tested in plane and solid geometry as
well as differential and integral calculus. Military Academy: 112; Harvard 98.
The contest was expanded in 1938 to a national event and the Putnam Competition
continues to this day as the most prestigious contest for mathematics undergraduates.
Alas, West Point has not won since. It is true the
exam covers the whole range of
undergraduate mathematics and until cadets were allowed to major in mathematics it
was difficult to compete
effectively. Of late President Lowell has had the last laugh: not
so much as an individual or team honorable mention for West Point as least as far
back
at 2000. Harvard has had hundreds of good individual scores. The Harvard teams have
finished as follows: 2009 (2); 2008 (1); 2007 (1); 2006(2);
2005 (1); 2004 (honorable
mention); 2003 (2); 2002 (1); 2001 (1) and 2000 (3). Update
for Harvard: 2010 (3); 2011 (1); 2012 (2); 2013(4); 2014 (2); 2015
(5); 2016(3); 2017(2); 2018 (1); 2019 (2);
2020 (COVID); 2021(3);
and 2022(2). No mention of West Point.