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Investigating Euler's Polyhedral Formula Using Original Sources - Euler's Grand Problem

Lee Stemkoski (Adelphi University)

 We would be remiss if we did not mention the grand finale to this wonderful work of Euler. In paragraph 59, Euler presents a grand problem: to classify all solids.

Figure 13. Euler's Grand Problem.

This requires results from throughout the paper. This is a good final experience for students as well.

Thanks to the groundwork laid by Euler, this is a relatively straightforward question. First, he starts by choosing a value for V. Using the inequalities previously developed, there is a range of possible values for F. For each possible value of F, use the polyhedral formula (Proposition 4) to find the corresponding value of E. Euler lists the possibilities in a table, for solids with up to 10 vertices, along with their names using his nomenclature.

Figure 14. Partial Solution to the Grand Problem.

This naturally lends itself to a number of activities:

  • Create or extend the table given by Euler.
  • Draw diagrams of the solids mentioned in the table. The shape of the faces of a given solid—triangular, square, etc.—is not obvious, given the tabulated data.
  • Match Euler's names with modern names. For example, among the first seven entries in this table, you will find what we refer to today as a tetrahedron, a square pyramid, a pentagonal pyramid, a triangular dipyramid, an octahedron, a triangular prism, and one solid without a standard name.

This concludes our overview of the many interesting results to be found in this paper, and we hope we have inspired the reader with some ideas for using this original paper of Euler in the classroom.

In closing, we paraphrase a famous saying:

Read Euler, Read Euler; he is an author for us all . . . even our students.


Lee Stemkoski (Adelphi University), "Investigating Euler's Polyhedral Formula Using Original Sources - Euler's Grand Problem," Convergence (April 2010), DOI:10.4169/loci003297