Chapter 13, Problem 77E

During the time of the Greeks the star pentagram shown in Fig. 13-42 was a symbol of the Brotherhood of Pythagoras. Consider the three segments of lengths x, y and z shown in the figure.

Figure 13-42

a. Show that $x/y=\text{φ}$, $\left(x+y\right)/z=\text{φ}$ and $\left(x+y+z\right)/\left(x+y\right)=\text{φ}$.

b. Show that if $y=1$, then $x=\text{φ}$, $\left(x+y\right)={\text{φ}}^2$ and $x+y+z={\text{φ}}^3$.