The golden ratio is a mysterious irrational number that appears in mathematics, nature and the arts.

The history of the golden ratio begins with Euclid’s comment in “Elements” about a point divides a line into an “extreme and mean ratio.” Afterward intellectuals began studying this ratio and found several links to other mathematical aspects such as the ratio of successive Fibonacci numbers approaches the value of the golden ratio. In geometry, the golden ratio appears frequently in pentagrams and pentagons. When the vertices of a five-sided pentagon are connected a star is formed and the ratios of various segments in the star all are equivalent to the golden ratio. Furthermore, in chemistry the golden ratio appears in Buckminsterfullerene also known as “buckyball,” which is an important crystalline allotrope of carbon (the other two are diamond and graphite.) Buckyball composes of 60 carbon atoms bonded together to form a soccer-ball arrangement and the coordinate system uses the golden ratio.

In computation analysis, the golden ratio appears as the rate of convergence for the secant method. Furthermore, there is an algorithm that uses the golden ratio to find the maximum or minimum of a function.

In addition, the golden ratio plays an extensive role in nature and the arts. Known as an “aesthetically pleasing” ratio, the golden ratio appears in music, architecture and paintings. One of the most famous examples of the golden ratio is found in the great pyramids in Egypt. Leonardo da Vinci studied the golden ratio and it is assumed that several of his paintings including the “The Last Supper” incorporated the golden ratio.

Currently, the golden ratio is often used an appealing factor for publicity. In Newton, Massachusetts, the Hotel Indigo claims to utilize the golden ratio throughout the building such as in the headboard of the beds. Even diets are created from the golden ratio, the ‘Da Vinci Diet’ created by Stephen Lanzalott claims to use a “complicated” formula to break down ratios of carbohydrates, protein and fat. There are also several disputed claims of the golden ratio such as the sea shells of the chambered nautilus. While many mathematicians argue that the spiral directly relates to Fibonacci’s spiral, there are skeptics that claim that the spiral is purely logarithmic and there is no evidence supporting why the shells would develop into a golden ratio spiral. Overall, the golden ratio is a fascinating complex number and will continue to appear in today’s society.

Sources:

http://goldennumber.net/

http://en.wikipedia.org/wiki/Golden_ratio

http://www.cbsnews.com/stories/2004/06/15/health/main623260.shtml

http://www.sciencenews.org/articles/20050402/mathtrek.asp