The Golden ratio is approximated to be equal to 1.618033989. According to golden rule two quantities are said to be in a golden ratio if the ratio of larger quantity to smaller one is equal to ratio of the sum of larger quantity and the smaller one i.e. a+ba = ab = 1.618033989, where, a’ is the larger quantity and b is the smaller one. This ratio is also called the golden number, divine ratio, the golden section or even the golden mean. The concept of golden rule is attributed to Pythagoras and his followers. The oldest definition of golden rule can which made use of the words ‘extreme and mean ratio’ is found in Euclid’s Elements. The golden ratio was studied by ancient Greek mathematicians because it frequently appeared in geometry. For example, the division of line into in extreme and mean ratio is an important concept in geometry of regular pentagons. Over years golden ratio properties has captured imagination of professionals from different fields such as architect, artist among others. These professionals proportionate their work to in accordance to approximation of the golden ratio. This ratio can also be used in relation to Fibonacci numbers (Webber, & Barnes, 1994).
The Fibonacci numbers are also called Fibonacci series were invented by a person called Leonardo born in Pisa Italy. This series is either represented as 0, 1, 1, 2, 3,5 ,8, 13, 21, 34, 55, 89, 144 or even as 1,1, 2, 3,5 ,8, 13, 21, 34, 55, 89, 144. Therefore any term in the series can be obtained by the formulae FN = Fn-1 + Fn-2 where n represents any term in the series. This numbers have a close relationship with golden ratio because the closest rational approximation is 1/1 =1, 2/1=2, 3/2= 1.5, 5/3= 1.66, 8/5=1.6, 13/8= 1.625..this shows that as the series progresses the ratio between consecutive terms converge towards the value of golden ratio i.e. 1.618033989. However, the ratio can never be equal to the exact value of the golden rule (Webber, & Barnes, 1994).
This numbers series are used in Fibonacci search technique, Fibonacci cubes which are used for interconnecting parallel and distribution systems and Fibonacci data structure. Fibonacci numbers naturally appears in the way: leaves arrange themselves on a stem, arrangement of a pine cone, and the fruit sprout of a pineapple (sPosamentier, & Lehmann, 2007).
References
Posamentier, A. S., & Lehmann, I. 2007. The fabulous Fibonacci numbers. Amherst, N.Y.: Prometheus Books.
Webber, B., & Barnes, T. 1994. How to be brilliant at algebra. Bedfordshire, UK: Brilliant Publications.
