Golden ratio

(PD) Diagram: Joel Holdsworth
demonstrating a simple method to construct a Golden Ratio rectangle

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

The golden ratio 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.

Johannes Kepler took considerable interest in the subject, and in his 1596 book, Mysterium Cosmographicum, wrote:

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold, the second we may name a precious jewel.

Kepler seems to have particularly associated this ratio with forms of plants, and all organic growth. In more recent years Jay Hambridge wrote several books devoted entirely to the Golden ratio, pointing out many instances of its presence on organic forms, as well as in the art and architecture of Attic Greece.

It has been claimed that the golden ratio is found in the proportions of architectural structures ranging from the Great Pyramid of Giza, to the Parthenon, to Chartres Cathedral. In the 20th century, the golden section was applied by the French architect Le Corbusier, who used the proportion in the design of virtually all his architectural works, and in his his painting, and who wrote a series of books on the subject called The Modulor.

To 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 $\scriptstyle \Phi ={\frac {a}{b}}={\frac {a+b}{a}}$.

Properties

Rewriting this definition gives

$\Phi ={\frac {a}{b}}={\frac {a+b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\Phi }}$,

which leads to

$\Phi ^{2}-\Phi -1=0$,

a quadratic equation wit the solutions

$\Phi ={\frac {1+{\sqrt {5}}}{2}}=1+{\frac {1}{\Phi }}=1{,}618\,033\,988\dots$ .

and

${\bar {\Phi }}={\frac {1-{\sqrt {5}}}{2}}=1-\Phi =-{\frac {1}{\Phi }}=-0{,}618\,033\,988\dots$

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.