Thursday, 17 March 2011

Trisecting an Angle

Just down the road about an hours drive  from my school here in Lakenheath in the East Anglia region of England is the pretty little village of Woodbridge.  Until the 90's, it too had a US Dept of Defense High School. 
Situated on the banks of the River Deben, Woodbridge is renowned for its sailing, riverside walks, pubs, restaurants, independent shops and theatre.  It has a high-street, the British version of a "main street" in the USA, with a number of pretty little shops.  In 1860, one of the shops along the high street was a china shop owned and operated by Joseph Roberts Morley.  Joseph's son Frank was a bright young man who would have gone on to be a life long and productive resident perhaps, except by chance he developed a love of chess, and found himself to be very good at it.  This interest led him to meet another chess aficionado, who had spent some of his formative years not far away on his uncle's farm in Suffolk.  George Biddell Airy, the Astronomer Royal, recognized young Frank's chess skills and his potential in mathematics, and had an influence on getting him to compete for a scholarship to Kings College in Cambridge, which Frank won.  

Morley went on to graduate from Cambridge in 1884 and seemed to have not overly impressed anyone there with his mathematical power.  Like thousands of mediocre graduates before and after, he went into teaching.  As a teacher Morley would prove to be exceptionally capable.  After he moved to the US, first at Haverford College , then to St. Johns, where he supervised 48 graduate students.  His research seems not to have been on the mainstream of mathematics, but he is remembered for his ability to "have on hand a sufficient variety of thesis problems to accommodate particular tastes and capacities."  (as another connection to my interest in Morley,and to my present school, one of my ex-calculus students will graduate from Haverford this year and one graduated from St. Johns last year)


But Morley is remembered most today for a singular theorem which bears his name in recreational literature.  Simply stated, Morley's Theorem says that if the angles at the vertices of any triangle (A, B, and C in the figure) are trisected, then the points where the trisectors from adjacent vertices intersect (D, E, and F) will form an equilateral triangle.
   In 1899 he observed the relationship described above, but could find no proof. It spread from discussions with his friends to become an item of mathematical gossip. Finally in 1909 a trigonometric solution was discovered by M. Satyanarayana. Later an elementary proof was developed. Today the preferred proof is to begin with the result and work backward. Start with an equilateral triangle and show that the vertices are the intersection of the trisectors of a triangle with any given set of angles. For those interested in seeing the proof, check Coxeter's Introduction to Geometry, Vol 2, pages 24-25.


About 1796, the nineteen year old Karl Gauss proved that certain regular polygons could be created with only the classic construction tools of straight edge and compass. Some forty years later, Wantzel completed this work by showing that only the regular polygons Gauss had described could be so constructed. As a consequence of this, it was finally proven that a general angle could not be trisected with straight edge and compass, thus ending the search for one of the classic problems of antiquity. From then until now, mathematicians and math professors have been beset by "proofs" from zealous amateurs who have triseceted an angle.  H S M Coxeter has suggested that from this time on people felt uneasy about the mention of trisecting an angle. This, he thinks, probably contributed to the reason that Morley's Theorem was not discovered until near the dawning of the 20th century. 

I was impressed that all three of Morley's sons went on to become Rhodes Scholars.  Felix was the editor of the Washington Post, a Pulitzer Prize winner, and then President of Haverford College.  Another, Christopher, became e a novelist, most remembered for " Kitty Foyle" .  A third, Frank V, was a director of the publishing firm Faber and Faber he also was a mathematician and worked with his father. It was Frank who wrote of his father,
.. then he would begin to fiddle in his waistcoat pocket for a stub of pencil perhaps two inches long, and there would be a certain amount of scrabbling in a side pocket for an old envelope, and then there would be silence for a long time; until he would get up a little stealthily and make his way toward his study - but the boards in the hall always creaked, and my mother would call out, "Frank, you're not going to work!" - and the answer would always be, "A little, not much!" - and the study door would close.
(It wasn't hard to gather that my father was working at geometry, and I knew pretty well what geometry was, because for a long time I had been drawing triangles and things; but when you examined the envelope he left behind, what was really mysterious was that there was hardly ever a drawing on it, but just a lot of calculations in Greek letters. Geometry without pictures I found it hard to approve; indeed, I prefer it with pictures to this present day.)
 The senior Morley continued to be an impressive chess player and once defeated the algebraist and world chess champion,  Emmanuel Lasker.

I was reminded of Morley and his theorem recently when
To conclude, one of the sixty-plus problems that he posted to the Educational Times:

Show that on a chess-board the number of squares visible is 204, and the number of rectangles (including squares) visible is 1296; and that, on a similar board with n squares in each side, the number of squares is the sum of the first n square numbers, and the number of rectangles (including squares) is the sum of the first n cube numbers.






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