Chapter 3.5, Problem 45PPSE

Solution

Tocalculate:The numbers of vertices, edges and faces if every face has five edges and every edge is shared by two faces. Every face has five vertices and every vertex is shared by three faces. The sum of the number of vertices and faces is two more than the number of edges.

The number of vertices are 20, edges are 30 and faces are12.

Given information:

It is given that every face has five edges and every edge is shared by two faces. Every face has five vertices and every vertex is shared by three faces. The sum of the number of vertices and faces is two more than the number of edges.

Calculation:

Consider the given information.

Let x , y and z are the number of edges, faces and vertices respectively.

Since every face has five edges so this can be written as $x=5y$ .

Also, every edge is shared by two faces so $x=\frac{5}{2}y$ .

Every face has five vertices and every vertex is shared by three faces, so $3z=5y$ .

  $z=\frac{5y}{3}$

The sum of the number of vertices and faces is two more than the number of edges.

  $\begin{array}{l}z+y=x+2\\ z+y-x=2\end{array}$

Substitute the respective values into the above equation.

  $\begin{array}{c}\frac{5}{3}y+y-\frac{5}{2}y=2\\ 10y+6y-15y=12\\ y=12\end{array}$

Put $y=12$ into $x=\frac{5}{2}y$ .

  $\begin{array}{c}x=\frac{5}{2}\left(12\right)\\ x=30\end{array}$

Put $y=12$ into $z=\frac{5y}{3}$ .

  $\begin{array}{c}z=\frac{5\left(12\right)}{3}\\ z=20\end{array}$

The number of vertices are 20, edges are 30 and faces are 12.