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In H. Eves, Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Investigating Euler's Polyhedral Formula Using Original Sources
Euler's Grand Problem
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:
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:
Stemkoski, Lee, "Investigating Euler's Polyhedral Formula Using Original Sources," Loci (April 2009), DOI: 10.4169/loci003297
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