The Golden Ratio and Fibonacci sequence
The Golden Ratio and Fibonacci sequence
The Golden Ratio: The golden ratio is a specific, irrational constant represented by the symbol “Phi” which may be viewed below. The “phi” symbol represents the golden ratio which is a number equal to 1.618 and it is a constant that occurs in numerous mathematical and design areas such as art, architecture and geometry (Dunlap 17). The golden ratio is alternatively known as the golden mean, medial section, golden cut, divine section, mean ratio, mean of Phidias and the golden number. The constant appears in numerous areas and is thus a true representative of some constant that is seemingly present in many areas within nature. The idea behind the constant may be illustrated by the simple act of dividing a line into two parts where there is one longer and another shorter part (Dunlap 29). For the division to count as a golden ratio there is a need for the division of the longer part by the smaller part to yield the value of the whole part divided by the longer part of the piece.
The acquired number is 1.61803398874989484820…and it may continue into infinity because it has the properties of an irrational number. The calculation of the golden ratio may take the following formulae and steps of calculation. Firstly, the number is divided by one (1/ designated number). Thereafter, the obtained number is added by one unit and the process may start from the first step. The process is supposed to go on till the number draws closer and closer to the golden ratio (Livio 56). Generally, in the arts and mathematics any two quantities may be in a golden ratio when the ratio of the added up quantities to a large amount of quantity is equivalent to a ratio that is obtained when the larger amount is divided by the smaller amount in the ratio. This results in an irrational number or constant which is estimated to be 1.6180339887.
The Fibonacci sequence
The Fibonacci sequence is a form of mathematical discovery made by Leonardo Pisano Bigollo. Leonardo was a famous mathematician that was well renowned in his times. Leonardo is well known for the spreading of the Hindu-Arabic numerals in some parts of Europe (Livio 71). The ratio came to be known primarily through his published work in the 13th century, titled the “Book of Calculation” or ‘Liber Abaci.’ In actual sense Leonardo did not discover the Fibonacci numbers, but he used them as an example in his published work-Liber Abaci-and they were named after him thereafter. Leonardo’s work posed a problem and sought its solution. The problem involved determining how the population of rabbits grew based on ideal assumptions.
The solution to this problem presented a generation-by-generation number sequences that would later be known as the Fibonacci numbers. These number sequences were earlier known by Indians in the 6th century, but it was through Fibonacci’s work on the “Book of calculation” that these became known in the western world (Dunlap 107). The Fibonacci sequence has a series of numbers in which the every number in the sequence is a sum of the previous two and the series starts with 0 and 1. The Fibonacci sequence may be represented as follows. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987….
The relation between the Golden ratio and Fibonacci numbers:
As one moves higher in the Fibonacci sequence, two consecutive numbers in the sequence tend to have a division result that approaches the golden ratio by an estimated 0.618:1 or 1: 1.618. The Fibonacci sequence and the golden ratio appear to be two different and somewhat unrelated topics, but interestingly they produce the same number. The fact that the obtained number-the golden ratio-is not rational implies that this is not a mere coincidence. Therefore, any two numbers in the Fibonacci sequence taken one after the other produce a number closer to the golden ratio and in fact as one goes higher in the sequence the closer the obtained results near the golden ratio. The same applies when one picks two whole numbers to start the sequence (Livio 83).
The application of the Golden ratio
The golden ratio is applied in beauty and architecture and quite a number of artists believe that the ratio makes very beautiful shapes in art and architecture. The architects have taken up the ratio and applied it on most architectural structures and the ratio can be identified in a number of great architectural buildings including the Greece Parthenon. Luca Pacioli a Franciscan friar recommended the use of the golden ratio in obtaining pleasing and harmonious proportions. The ratio’s application in most forms of art brought about a harmonious, proportional look to the art (Schwach, Polanski & Blacker 1). Other famous examples of the application of the golden ratio include the Mosque of Kairouan which shows a consistent pattern of the golden ratio in the entire design of the mosque including the dimensions of the praying space, the overall mosque proportion, the minaret and the mosque’s court. The evidence from these ancient architectural monuments shows that the golden ratio has been in use for a very long time in construction of various religious and aesthetic structures and the ratio has been a tool to enhance the aesthetical appeal of the work. According to Le Corbusier, a Swiss architect the universe was ordered mathematically and the golden ratio is what bound the universe whereas, the Fibonacci series provides the rhythm of the eye which results in the harmony that is visually perceived (Schwach et al. 1)
The golden ratio has also been applied in painting by famous artists such as Salvador Dali. Influenced by Matila Ghyka’s works Salvador applied the golden ratio to the production of one of his master pieces known as “The Sacrament of the last supper.”Perceptual studies have also shown that the golden ratio may be responsible for the human perception of beauty and harmony, though most psychological researches in this direction have been inconclusive (Schwach et al. 1). The golden ratio’s significance and appearance in art may be probably explained by the fact that it occurs frequently in most geometrical works. The ratio turns up in figures that have pentagonal symmetry (for example the diagonal of pentagon has a length that is the golden ratio’s multiple of its side and the icosahedron’s vertices are equated to three mutually orthogonal golden triangles) (Livio 92). This look into the golden ratio and the Fibonacci sequence implies that these are representations of numbers in nature and portrayals of how mathematics is presented within natural set ups. The ratio seems to define shape whereas; the sequence offers rhythm to the eye in the observance of nature.
Works Cited
Dunlap, A. R. The golden ratio and Fibonacci numbers. World Scientific, 1997. Print
Livio, Mario. The golden ratio: the story of phi, the world’s most astonishing number,Broadway Books, 2003. Print
Schwach, M. Polanski, J. and Blacker, S. The golden ratio retrieved on 14th June, 2011 from http://www.geom.uiuc.edu/~demo5337/s97b/, 2011. Web
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