OF THE GOLDEN RATIO
This paper explores the applications of the golden ratio in different aspects of civilization such as art, engineering, architecture, nature and science.
The golden ratio (represented by the Greek letter φ) is also referred to as the divine ratio, golden section, and Phi. It is the ratio that can be described by the golden rectangle (See Figure 1). The golden rectangle has sides which measures x and x+1 respectively. The ratio (x+1)/x = x/1. Rearranging the expression to quadratic equation form, this becomes x2-x-1. Solving the equation yields two irrational numbers. The two solutions are (1+5)/2 [approximately 1.618] and (1-5)/2 [approximately 0.618]. The first solution, roughly 1.618, is referred to as the golden ratio or the Phi.
Figure 1: The Golden Rectangle
The golden ratio, Phi, can be seen in architecture, art, nature, engineering, and science. Even some modern logo designs are attributed to be based on Phi. The Greek symbol Phi (φ) is in honour of the Greek sculptor, Phidias. Phidias designed the Parthenon in Athens and the statue of Zeus at Olympia. The use of Phi in antiquity and modern science has been widely observed and critiqued. According to a research by Fett (2006), this ratio can be observed in the artworks and architecture of the Greeks, Egyptians and even the Renaissance thinkers such as Leonardo da Vinci. There were also contemporary scholars who refuted this notion such as Markowsky (1992) and Falbo (2005). However, the latter authors are concerned more with the numerical accuracy of the use of Phi such as using ±2% allowance. In this paper, applications with measurements approximating the golden ratio will be discussed in the light of these three scholarly articles.
In architecture, the Parthenon built by Phidias uses Phi. The Parthenon can be inscribed in a golden rectangle. The ratio of the base to the height is approximately 1.618. Furthermore, the statue of the Greek goddess Athena inside the Partheneon also had the proportions of the golden ratio. Therefore, the human body where sculptures were based also have the golden ratios in it. For instance, the ratio of the human height to the distance from the head to the navel is about 1.6. According to Fett (2006), this proportionality between the ratios of the human body and architecture of temples is prevalent in ancient Greece. Another architecture using Phi is the Great pyramids of Cheops in Egypt. The ratio of the base length to the height is approximately 8/5 or 1.6.
Some examples in nature are the pinecones and the sunflowers (Elam, 2001). The seeds of the pinecones and sunflowers are parts of a spiral growth pattern. Their design approximate logarithmic spirals of the equation r = becθ. The ratio b is approximately equal to Phi. This spiral pattern can also be seen in animals such as in the tusks of elephants and horns of rams. The helix structure of the DNA is also based on the golden ratio.
For art, the famous Renaissance painters use a method called rabatment. In rabatment, the painters use diagonals and squares to divide the area they are painting. Quite interestingly, the resulting ratios of the spaces are golden ratios. Examples of these paintings are Madonna in Glory by Bondone, Mona Lisa by Leonardo da Vinci, and Santa Trinita Madonna by de Pepo. In modern times, logos of famous companies such as Apple, Pepsi and Toyota are also based on the golden ratio (Banerjee, 2011).
The golden ratio’s applications in the real world are very evident not only in history and nature, but also in science and modern art. Phi covers different aspects of our civilization. Even at modern times, its aesthetic appeal to people has prompted researchers to analyze data further and creating patterns from it.
Based on this short study, the researcher has better factual view of the golden ratio in terms of its description and derivation. The researcher also understood more the value of critiquing. Some authors critic that the ratio is not approximately equal to 1.618 for the chambered Nautilus, or even in the literature such as Virgil’s Aeneid (Markowsky, 1992). The numerical data they provide also prove this. However, there are still some other varied applications which use Phi both in the past and present. Particular focus on approximation instead of accuracy is emphasized on the discussion of the real world applications of Phi.
Reference
Banerjee, S. (July 29, 2011). Golden ratio in logo designs. Retrieved from http://www.banskt.com/blog/golden-ratio-in-logo-designs/
Elam, K. (2001). Geometry of design: Studies in proportion and composition. New York: Princeton Architectural Press.
Falbo, C. (2005). The golden ratio- a contrary viewpoint. The College Mathematics Journal. http://www.jstor.org/stable/30044835
Fett, B. (2006). An in-depth investigation of the divine ratio. The Montana Council of Teachers of Mathematics. Retrieved from http://www.math.umt.edu/tmme/vol3no2/tmmevol3no2_montana_pp157_175.pdf
Markowsky, G (1992). Misconceptions about the golden ratio. The College Mathematics Journal. Retrieved from http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf
