golden ratio

The “Golden Ratio”, or $\varphi$, has the value

$$1.61803398874989484820\mathrm{\dots }$$

This number gets its rather illustrious name from the fact that the Greeks thought that a rectangle with ratio of side lengths equal to $\varphi$ was the most pleasing to the eye, and much of classical Greek architecture is based on this premise. In , an aesthetically pleasing aspect of a rectangle with this ratio, from a mathematical viewpoint, is that if we embed and remove a $w\times w$ square in the below diagram, the remaining rectangle also has a width-to-length ratio of $\varphi$.

Above: The golden rectangle; $l/w=\varphi$.

$\varphi$ has plenty of interesting mathematical , however. Its value is exactly

$$\frac{1+\sqrt{5}}{2}$$

The value

$$\frac{1-\sqrt{5}}{2}$$

is often called ${\varphi }^{\prime }$. $\varphi$ and ${\varphi }^{\prime }$ are the two roots of the recurrence relation given by the Fibonacci sequence. The following hold for $\varphi$ and ${\varphi }^{\prime }$ :

• $\frac{1}{\varphi }=-{\varphi }^{\prime }$

• $1-\varphi ={\varphi }^{\prime }$

• $\frac{1}{{\varphi }^{\prime }}=-\varphi$

• $1-{\varphi }^{\prime }=\varphi$

and so on. These give us

$${\varphi }^{-1}+{\varphi }^0={\varphi }^1$$

which implies

$${\varphi }^{n-1}+{\varphi }^n={\varphi }^{n+1}$$