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Perhaps no other number has fascinated scientists, mathematicians and artists alike as the Golden Ratio. Also known as the golden mean or golden section (depending on its application), the Golden Ratio is observed by many people to occur naturally, manifesting itself in many natural shapes and geometry. This interesting phenomenon has resulted in an almost cult-like fascination among many people. In fact, there are people, including notable scientists and mathematicians with the likes of Johannes Kepler, Leonardo Da Vinci and Fra Luca Pacioli, who refer to the Golden Ratio as the ‘Divine Proportion,’ suggesting that the Golden Ratio have mythical and mysterious properties (Rose 39). One of the earliest manifestation of the Golden Ratio can be observed in the works of the Greek sculptor, Phidias, who is attributed for designing the architecture of the Parthenon, which is believed to have been built sometime in the 5th century B.C. (Livio 5). The Greek sculptor, Phidias, is often considered as the one who discovered the Golden Ratio. However, Phidias is just one of the many who have used the Golden Ratio in their works. This misconception regarding Phidias’ being the pioneer in the discovery and utilization of the Golden Ratio can be attributed to the 1909 suggestion of the American mathematician, Mark Barr, to designate the Greek symbol phi (ϕ) to the Golden Ratio in honor or Phidias (Rose 39). The Golden Ratio, however, must have been discovered much earlier. Despite the fact that the discovery of the Golden Ratio could not be accurately determined, it is believed that the Pythagoreans were already aware of the Golden Ratio as early as the 6th century B.C. based on the geometry and symbols that they use. As observed by one scholar, “the Pythagorean preoccupation with the pentagram and the pentagon, coupled with the actual geometrical knowledge in the middle of the fifth century B.C., make it very plausible that the Pythagoreans, and in particular perhaps Hippasus of Metapontum, discovered the Golden Ratio and, through it, incommensurability” (Livio 35).
One of the most famous example of the application of the golden ratio in classical architecture can be observed in the Parthenon. The use of the golden ratio in the Parthenon, which is believed to have been designed by the Greek sculptor and architect, Phidias, is a well-known story. Some scholars, however, refute the use of the golden ratio in the Parthenon structure. The mathematician, George Markowsky of the University of Michigan, for instance, deny the presence of the golden ratio in the building’s architecture and believe that any manifestation of such ratio is only accidental and not planned. A closer inspection of the Parthenon’s architecture would also reveal that it does not exactly represent the golden ratio in its dimensions. According to one scholar, “Parthenon's architects might have decided to base its design on some prevailing notion for a canon for aesthetics. However, this is far less certain than many books would like us to believe and is not particularly well supported by the actual dimensions of the Parthenon” (Livio 75). The Greeks, during the time of Phidias, however, are already well aware of the golden ratio because of the Pythagoreans. According to one scholar, the concept of aesthetics in classical Greek architecture is also equivalent to harmony, which is also associated with the golden proportions (Stakhov 1125). In this regards, it can be deduced that the Parthenon’s design is, in fact, based on the golden ratio.
Despite its manifestations in the works of the Pythagoreans and the ancient Greeks, a precise definition of the Golden Ratio, did not emerge until 300 B.C., when it was geometrically demonstrated by Euclid in his famous work, ‘Elements’ (The Ohio State University 2). Euclid called the golden ratio as the ‘extreme and mean ratio’ and defined it as the proportion obtained when a straight line is cut in such a way that the greater segment is proportional to the whole line in the same way that the smaller segment is proportional to the greater segment as shown in the figure below.
Figure 1. Euclid’s representation of the golden ratio (Cornell University).
Mathematically, this ratio is determined to be a non-repeating, non-terminating number and can be expressed as:
ϕ = 1+52= 1.618033989
The Golden Ratio does not only apply to lines, but also to geometric shapes, which follows the same principle. A rectangle and a triangle, for example, can be considered as ‘golden’ or having a golden section if the ratio of its sides corresponds to the proportion described by Euclid. Similarly, geometric shapes such as the pentagon and inscribed star as well as the Icosahedron and the Dodecahedron are observed to possess golden ratio properties. Another common manifestation of the golden ratio can also be observed in naturally occurring spirals such as the nautilus shell. It can be observed that a similar spiral can be made if a golden rectangle is split into smaller squares and golden rectangles and a curve is traced as shown in the figure below.
Figure 2. The Golden Spiral (Cornell University).
An interesting relationship can also be observed between the golden ratio and the Fibonacci series. The Fibonacci series was discovered by the Italian mathematician, Leonardo of Pisa or Leonardo Fibonacci. The series first appeared on his work ‘Liber Abacci,’ that he wrote in 1202, which can be obtained when one considers an innocent looking question regarding rabbit productivity. According to Fibonacci, “A certain man puts a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” (Benavoli, Chisci, and Farina 7). An attempt to answer the question would reveal a sequence of number known as the Fibonacci series in which each number is the sum of the preceding two numbers. The first ten numbers of the series in this particular rabbit problem, for instance, is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. The 12th month is found using this general mathematical equation, Fn = Fn−1 + Fn−2 ; where Fn is the nth Fibonacci number and Fn−1 and Fn−2 are the two preceding numbers respectively. It should be noted, however, that as the Fibonacci number increases, its ratio relative to Fn−1 also approaches the Golden Ratio. Mathematically, this relationship can be expressed as:
ϕ≈FnFn−1
Using this equation, it can be found that the larger the nth Fibonacci number, the closer is its value to the golden ratio. The 18th, 19th and 20th Fibonacci number when plugged into the equation, for example, would result to:
ϕ18≈F18F18−1=25841597=1.6180338; ϕ19≈F19F19−1=41812584=1.618034;ϕ20≈F20F20−1=67654181=1.6180339
The relationship between the golden ratio and the Fibonacci series could not be dismissed primarily because of the many natural phenomena that follows this order. Leonardo da Vinci’s and Johannes Kepler, who is believed to have independently rediscovered the golden ratio by observing the sequence of leaf arrangement, for instance, are quite convinced that the ratio has something to do with the natural order and composition of things (Livio 110). The mysterious relationship between nature; the golden ratio; and the Fibonacci number had not yet been fully established. Even so, these numbers still fascinate modern society and is seen and applied in many mathematical, scientific, architectural and artistic works.
Works Cited
Benavoli, A., L. Chisci, and A. Farina. “Fibonacci Sequence, Golden Section, Kalman Filter and Optimal Control.” Web. 13 Jan. 2017.
Cornell University. “The Golden Ratio.” 2012. Web. 13 Jan. 2017.
Livio, Mario. The Golden Ratio, The Story of Phi. The World’s Most Astonishing Number. New York: Broadway Books. Web. 13 Jan. 2017.
Rose, Nicholas J. “The Golden Mean and Fibonacci Numbers.” 2014. Web. 13 Jan. 2017.
Stakhov, Alexey. “Fundamentals of a New Kind of Mathematics Based on the Golden Section.” 2005. Web. 13 Jan. 2017.
The Ohio State University. “The Golden Ratio and Fibonacci.” Web. 13 Jan. 2017.
