Golden ratio

PD Image
The old townhall in Leipzig. The tower is positioned such between the left (a) and right (b) sections that $\scriptstyle {\frac {a}{b}}$ equals the golden ratio.

The golden ratio, also frequently known by a number of other names such as golden section or golden mean, is a mathematical proportion that is important in the arts and interesting to mathematicians. In architecture and painting, some works have been proportioned to approximate the golden ratio ever since antiquity, when, supposedly, some of the buildings on the Acropolis derived their eye-pleasing esthetics from the use of this ratio in determining the length of the buildings to their height and width.

According to the Merriam-Webster's Collegiate Dictionary, Eleventh Edition, the proportion is derived from two segments in which "the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller."

To be more elaborate: if there is a longer line segment $\scriptstyle a\$ and and a shorter line segment $\scriptstyle b\$, and if the ratio between $\scriptstyle a+b\$ and $\scriptstyle a\$ is equal to the ratio between the line segment $\scriptstyle a\$ and $\scriptstyle b\$, this ratio is the golden ratio. The value of the golden ratio is $\scriptstyle \Phi ={\frac {a}{b}}={\frac {1+{\sqrt {5}}}{2}}=1{,}618033988\dots$

Properties

• If ${\frac {a}{b}}={\frac {1+{\sqrt {5}}}{2}}$  it follows that ${\frac {a}{b}}={\frac {a+b}{a}}=1+{\frac {b}{a}}$

With $\Phi =1+{\frac {1}{\Phi }}$ we could derive the infinite continued fraction of the golden ratio: $\Phi =1+{\frac {1}{\Phi }}=1+{\frac {1}{1+{\frac {1}{\Phi }}}}=1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{\Phi }}}}}}=\dots$

Thus

$\Phi \ =\ 1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+\dots }}}}}}$

• $\Phi \ =\ \lim _{n\to \infty }{\frac {F(n+1)}{F(n)}}$,

where $\ F_{n}$ is the n-th term of the Fibonacci sequence.