Chapter 12.4, Problem 3BE

a.

To determine

To find : if Euler’s formula is satisfied for the given condition.

Answer to Problem 3BE

The Euler’s formula is not equal to 2 for the given condition and this tell that the structure does not only have hexagon it may have another polygon.

Explanation of Solution

Given information :

The framework has n faces all hexagon thus, $F=n$ also number of vertices $V=\frac{6n}{3}$ and the number of edges is $E=\frac{6n}{2}$ .

Calculation :

Euler’s formula is given by

  $F+V-E=2$

Here $F=n$ , $V=\frac{6n}{3}$ and $E=\frac{6n}{2}$ .

Now check if Euler’s formula is satisfied for the given condition.

  $\begin{array}{l}F+V-E=n+\frac{6n}{3}-\frac{6n}{2}\\ F+V-E=n+2n-3n\\ F+V-E=0\end{array}$ .

The Euler’s formula is not satisfied.

This tells that the structure does not only have hexagon it may have another polygon.

Hence,

The Euler’s formula is not equal to 2 for the given condition and this tell that the structure does not only have hexagon it may have another polygon.

b.

To determine

To show : That $V=\frac{6n-12}{3}$ and $E=\frac{6n-12}{2}$

Answer to Problem 3BE

The Euler’s formula is satisfied.

Explanation of Solution

Given information :

Suppose that 12 of the n faces of the framework are pentagons.

Calculation :

Euler’s formula is given by

  $F+V-E=2$

Here, 12 is pentagons and $n-12$ are hexagons.

Therefore, number of vertices is

  $\begin{array}{l}V=\frac{6\left(n-2\right)}{3}\\ V=\frac{6n-12}{3}\end{array}$ and

Number of edges is

  $\begin{array}{l}E=\frac{6\left(n-2\right)}{2}\\ E=\frac{6n-12}{2}\end{array}$

Now check if Euler’s formula is satisfied for the given condition.

  $\begin{array}{l}F+V-E=n+\frac{6n-12}{3}-\frac{6n-12}{2}\\ F+V-E=n+\frac{2\left(6n-12\right)-3\left(6n-12\right)}{6}\\ F+V-E=n+\frac{12n-24-18n+36}{6}\\ F+V-E=n+\frac{12-6n}{6}\\ F+V-E=n+\frac{6\left(2-n\right)}{6}\\ F+V-E=n+2-n\\ F+V-E=2\end{array}$ .

Hence,

The Euler’s formula is satisfied.