Investigating Euler's Polyhedral Formula Using Original Sources

Editor's note: This article was published in April 2009.

Teachers of mathematics would like to motivate their students. Approaches to achieve motivation can be divided into three categories:

• You can use the present. How does a topic relate to material recently covered in the course?
• You can use the future. After learning a topic, you will be prepared to apply your knowledge and solve the many real-world problems you will undoubtedly encounter in the future.
• You can use the past. What is the historical context of a given topic? Why did anyone consider it in the first place? What was the original formulation and presentation like?

Let us assume that we wish to use the past.

Who should we study? If we are interested in examining original sources, a natural choice is Leonhard Euler. He is a master of exposition; he writes to be understood; he doesn't assume extensive background; he uses lots of examples; ideas gradually build on each other, and so on. In short, Euler writes in precisely the style that many of us hope to achieve when we teach.

What should we read? In the case of Euler, there are literally hundreds of things from which to choose! Perhaps we would like to choose a topic that is particularly "fun"; one that does not involve advanced topics such as calculus or mechanics. Even with this restriction, there is a variety of material available. Possible topics include: the bridges of Königsberg and the beginnings of graph theory, the 36 officers problem and Latin squares, knight's tours on a chessboard, investigating Fermat numbers, and many more. Here, we will discuss Euler's first paper on polyhedra. This topic is particularly appealing to a large audience of students because it is very visual. Particularly in mathematics, students tend to appreciate objects they can visualize and even manipulate—many students prefer the concrete (such as models you can construct) to the abstract (such as manipulations of functions and equations). However, there are plenty of functions and equations to be found in this topic; they are potentially easier to grasp because they relate to objects readily visualized.

For the Euler enthusiast, the particular paper described below is Elementa Doctrinae Solidorum, Eneström number 230 [Euler 1758].

Where can this paper be used? This topic is appropriate for a course in Discrete Mathematics, Graph Theory, History of Mathematics, or perhaps even a general course in mathematics for an audience of Liberal Arts students.

How can this paper be used and this module be effectively implemented? The basic material required is all available online. The Euler Archive (located at http://eulerarchive.maa.org/) has scanned versions of Euler's original publications freely available for download. The publication under discussion here may be downloaded in its entirety from the Euler Archive or via this link. In some cases, the Archive also includes links to translations and lists of references. The other major resource is a monthly column, How Euler Did It, written by Dr. Edward Sandifer, originally published at MAA Online and now available from the Euler Archive at http://eulerarchive.maa.org/hedi/index.html. The June 2004 and July 2004 columns also address the work of Euler discussed in this article; we will avoid overlapping the material discussed in these columns as much as possible. In addition to on-line resources, instructors may wish to compare and contrast Euler's results and presentation style with that of a modern textbook, such as [West 2001]. Furthermore, depending on the time available for this topic and the level of the course, [Lakatos 1976] is an excellent companion book that discusses Euler's polyhedral formula in depth.

This article concerns using Euler's original paper to investigate properties of polyhedra, in particular, convincing the reader that it is a worthwhile approach to take.