Counting Triangulations of a Convex Polygon

In a 1751 letter to Christian Goldbach (1690–1764), Leonhard Euler (1707–1783) discusses the problem of counting the number of triangulations of a convex polygon. Euler, one of the most prolific mathematicians of all times, and Goldbach, who was a Professor of Mathematics and historian at St. Petersburg and later served as a tutor for Tsar Peter II, carried out extensive correspondence, mostly on mathematical matters. In his letter, Euler provides a “guessed” method for computing the number of triangulations of a polygon that has n sides but does not provide a proof of his method. The method, if correct, leads to a formula for calculating the number of triangulations of an n-sided polygon which can be used to quickly calculate this number [1, p. 339-350] [2]. Later, Euler communicated this problem to the Hungarian mathematician Jan Andrej Segner (1704–1777). Segner, who spent most of his professional career in Germany (under the German name Johann Andreas von Segner), was the first Professor of Mathematics at the University of Göttingen, becoming the chair in 1735. Segner “solved” the problem by providing a proven correct method for computing the number of triangulations of a convex n-sided polygon using the number of triangulations for polygons with fewer than n sides [5]. However, this method did not establish the validity (or invalidity) of Euler’s guessed method. Segner communicated his result to Euler in 1756 and in his communication he also calculated the number of triangulations for the n-sided polygons for n = 1 . . . 20 [5]. Interestingly enough, he made simple arithmetical errors in calculating the number of triangulations for polygons with 15 and 20 sides. Euler corrected these mistakes and also calculated the number of triangulations for polygons with up to 25 sides. It turns out that with the corrections, Euler’s guessed method gives the right number triangulations of polygons with up to 25 sides. Was Euler’s guessed method correct? It looked like it was but there was no proof. The problem was posed as an open challenge to the mathematicians by Joseph Liouville (1809–1882) in the late 1830’s. He received solutions or purported solutions to the problem by many mathematicians (including one by Belgian mathematician Catalan which was correct but not so elegant), some of which were later published in the Liouville journal, one of the primary journals of mathematics at that time and for many decades. The most elegant of these solutions was communicated to him in a paper by Gabriel Lamé (1795-1870) in 1838. French mathematician, engineer and physicist Lamé was educated at the prestigious Ecole Polytéchnique and later at the Ecole des Mines[3, p. 601–602]. From 1832 to 1844 he served as the chair of physics at the Ecole Polytéchnique, and in 1843 joined the Paris Academy of Sciences in the geometry section. He contributed to the fields of differential geometry, number theory, thermodynamics and applied mathematics. Among his publications are textbooks in physics and papers on heat transfer, where he introduced the rather useful technique of curvilinear coordinates. In 1851 he was appointed Professor of Mathematical