# Investigating Euler's Polyhedral Formula Using Original Sources - Additional Results

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There are many, many more results in this paper of Euler than can be addressed in this article. We will briefly mention a few additional results.

Proposition 5 (paragraph 37) states:

It is not possible that $E + 6 > 3F$ or $E + 6 > 3V$.

Proposition 6 (paragraph 42) states:

It is not possible that $F + 4 > 2V$ or $V + 4 > 2F$.

Euler then combines many of his propositions in an exciting way. From Proposition 2, Corollary 2, we know that $2E ≥ 3F$, or equivalently, $\frac{3}{2}F ≤ E$. From Proposition 5, we know that it is not possible that $E + 6 > 3F$, or equivalently, $E ≤ 3F - 6$. Therefore, if we know the number of faces, $F$, of a polyhedron, we now know upper and lower bounds on the number of edges, $E$:

$\frac{3}{2}F ≤ E ≤ 3F - 6$.

From Proposition 6, we know that $F + 4 ≤ 2V$ and $V + 4 ≤ 2F$, or equivalently, that $\frac{1}{2}F + 2 ≤ V$ and $V ≤ 2F - 4$. Therefore, if we know the number of faces, $F$, of a polyhedron, we now know upper and lower bounds on the number of vertices, $V$:

$\frac{1}{2}F + 2 ≤ V ≤ 2F - 4$.

In Proposition 6, Corollary 5 (paragraph 47), Euler uses all of these formulas together and classifies all of the different possible solids by number of faces, vertices, and edges. He produces the following table:

Figure 12. Bounds on Vertices and Edges for Given Number of Faces.

In the first column are different possible values for the number of faces of a polyhedron ranging from $F = 4$ to $F = 25$. For each value of $F$ there is a range of possible values for $V$ and $E$, and these ranges are the contents of the second and third columns of the table, respectively.

This table can be used in a classroom lesson in a variety of ways. Students can be given the equations above to re-create the table for themselves or simply to extend the table (perhaps for solids with up to $30$ faces), or alternatively students could be given the table and asked to locate the equations in Euler's paper that are give rise to these values. Students could be asked to build a similar table: one that gives ranges for the values of $F$ and $V$ given values of $E$; this is Euler's Corollary 7 to this proposition (paragraph 49); Euler tackles the cases from $E = 6$ to $E = 60$.

Propositions 7, 8, and 9 are mentioned in Dr. Sandifer's column in How Euler Did It, and so we will not discuss them here.

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