Mathematics 101
The Golden Ratio and Its Use
The Golden Ratio (also known as the Golden Mean, the Golden Section, and the Divine Proportion) is the proportion of two quantities, such that their ratio is equal to the ratio of their sum to the biggest of those quantities. Due to its curious properties, throughout the history this ratio has been of interest not only to the mathematicians, but also to people of art – such as painters, architects, and designers.
The studies of this ratio began in ancient Greece, being related to the geometric problem of cutting a line section into the segments fitting under the Golden ratio definition. Denoting the mentioned segments by a and b, the following is true for the Golden Ratio to be observed: a+ba=ab. Substituting ab=φ yields the following equation: 1+1φ=φ. Here the Greek letter “Phi” denotes the value of the Golden Ratio, and by solving the equation one can arrive at its value of φ=5+12=1.6180339887 Only the positive root of the equation is taken into account, since the Golden Ratio relates to the proportion of the positive segments. In addition, the inverse of “Phi” is sometimes used under the same name of the Golden Section, denoted as “phi” (with a small letter), and equal to 1φ= 0.6180339887 , which is also the same as φ-1. Resultantly, an interesting property of “Phi” lies in that it is the only number, the square of which is bigger by 1 than the number itself.
According to Lidwell et al. (114), the Golden Ratio is commonly observed in the nature – namely, in the form of seashells, pine cones, and human body. For example, the wrist divides the hand and forearm in the proportion of the Golden Ratio; the navel divides the whole body in the proportion close to this ratio; and the lengths of the bones of a finger also represent the Golden Section proportion (Olsen 20). In a pine cone, the growth angle of the consecutive segments is close to the Golden Ratio (Dunlap 130). The study of the Nautilus pompilius mollusk shells show that the proportion of volumes of the neighboring chambers in a shell is also related to the Golden Section (Dunlap 135). In general, the Golden Ratio is argued to be inherent in the growth patterns of biological organisms (particularly those growing in the form of spirals).
The modern architecture examples, relating to the 20th century, include the works of the architects Erns Neufert and Le Corbusier, both of whom saw in the Golden Ratio the continuation of the great masters of the past, such as Vitruvius, Leonardo da Vinci's "Vitruvian Man", and the works of Leon Battista Alberti. It is claimed by Nikos Salingaros that the popularity of the Golden Ratio concept in this period was related to the rise of modernism and the “geometric fundamentalism”, which allowed the architect to use his intellect to appeal to the user’s senses, focusing on the mystical appeal of the irrational number “Phi”. Erns Neufert (1900-1986) was known for promoting the Golden Section idea as a key principle of his architectural works, due to the fact that of its being based on proportions of the human body, and, as such, also presenting the underlying link between all harmonies in architecture. He described this principle and its proposed applications in detail in his famous book on building design and planning Bauentwurfslehre (Architects' Data) in 1936 as an idea that could combine rational norming with aesthetic appeal (Frings 31). Similarly, Le Corbusier (1887–1965) describes the Golden Mean as the “natural rhythm, inborn to every human organism” (Frings 32), still, not strictly adhering to fixed proportions, but rather relating to “measure-ruler” concept. His Modulor system for the scale of architectural proportions, based primarily on the Golden Ratio, was applied in construction of Villa Stein in Garches in 1927, with the rectangular proportion of the building in ground plan and façade, as well as inner structure of the ground plan, being based on the Golden Mean proportion. An even more recent example, according to Salingaros, is the United Nations Secretariat Building in New York, which was designed in 1950 by Wallace Harrison and Max Abramovitz on the basis of earlier sketches by Le Corbusier. Its façade has the shape of an empty rectangular slab with proportion cited to have the ratio representing the Golden Mean.
However, the use of Golden Mean in art and design has also its opponents. First of all, as Salingaros argues, the building frequently cited as having Golden Ratio proportion are in fact rarely measured in practice. Moreover, the author states that experimental results have shown that people generally do not prefer the Golden Mean geometrical forms to other (not based on Golden Mean), and even cannot distinguish the Golden Ratio rectangles among other rectangles. Lidwell et al (114) advise to use the Golden Section in design only when other rules are unapplicable or impractical.
Therefore, dating back as early as the ancient Greeks, the Golden Ratio still remains one of the interesting mathematical concepts that still keeps fascinating people’s minds and has found its applications in art and design.
Works Cited
Dunlap, Richard. The golden ratio and Fibonacci numbers. Singapore: World Scientific Publishing, 2004. Print.
Frings, Marcus. "The Golden Section in Architectural Theory", Nexus Network Journal 4(1) Winter 2002: 9-32.
Knott, Ron. Fibonacci Numbers and the Golden Section. The Mathematics Department of the University of Surrey, UK, 1996. Web. 19 Apr. 2014.
Lidwell, William, Holden, Kritina, Butler, Jill, and Kimberly Elam. Universal principles of design: 125 ways to enhance usability, influence perception, increase appeal, make better design decisions, and teach through design. 2nd ed. Beverly, Mass.: Rockport Publishers, 2010. Print.
Olsen, Scott. The golden section: nature's greatest secret. New York: Walker, 2006. Print.
Salingaros, Nikos. Applications of the Golden Mean to Architecture. 22nd February 2012. Web. 19 Apr. 2014.
Solà-Soler, Jordi. Phi in the Great Pyramid. July 2012. Web. 19 Apr. 2014.
Van Mersbergen, Audrey. “Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic.” Communication Quarterly 46(2) 1998: 194-213. Print.
