Renae Lange History of Math 10/11/13 The Golden Ratio in Art and Architecture: A Critical Review As a mathematician, I experience a thrill in finding connections between mathematics and other disciplines. In fact, it is in these connections that math takes on meaning, application, and even beauty. The discovery of a mathematical concept played out in nature leads not only to a deeper understanding of that concept, but to a broader perspective of the world around us. There does, however, come a point when a perceived connection is stretched too far when information dabbles in being misleading or is falsely obtained. This may be the case with the mysterious or astonishing number known as phi or φ, which is commonly referred to as the golden ratio. It is believed that phi has uncanny appearances in every great discipline, ranging from art, music, architecture, theology, nature, and even the stock market! In this essay, I will provide a brief background on phi and its mathematical significance. Then I will address some of the mysterious claims between phi, art, and architecture. Last I will cite some mathematical objections to those claims. From biased and unbiased sources alike, the principle of a Golden Ratio is arises from the Greeks, particularly from a man named Euclid. The golden ratio occurs when a line segment is divided into two pieces with the following characteristics: that the ratio of the whole
segment to the larger piece is equal to the ratio of the larger piece to the smaller piece. This was called division in extreme and mean ratio (Markowsky, 1992, p. 2). While the term golden ratio is not directly attributed to Euclid, his idea of this division has led to an irrational number that has puzzled scholars for centuries. It rears its head in uncanny places, from the artistic to even other areas of math, such as the Fibonacci numbers. According to this ratio, phi is approximately equal to 1.61803398875. Like pi, we do not know all of the digits of phi. The golden ratio appears in many geometric constructions such as the ratio of side to base in a 72-72 -36 isosceles triangle (Markowsky, 1992 p. 3). A rectangle whose dimensions are a and b where a b and a/ b is considered a golden rectangle (Markowsky, 1992, p.3). Phi enthusiasts would claim that this golden rectangle plays a major role in the intricate pieces of Leonardo Da Vinci and other great artists. This being said, it is important to dive deeper into the mysterious claims of phi in art and architecture. To discover the supposed claims, I set out on a Google search for golden ratio in art and architecture. What I found was a gold mine a gold mine with the web address: www.goldennumber.net. Gary Meisner, the creator of the website, finds his inspiration for uncovering the mysteries of phi related back to his personal faith in God. He does not claim to be a mathematician just a man that loves math and is looking for a designer s signature through phi in the world around us. He, like me, believes that there is an intelligent designer that created the world. We can see marks of this intelligent God through His handiwork in nature particularly in its mathematical patterns. I appreciate this mindset of looking for God in the design of life. However, from what I have encountered of his website, I have reason to
question his claims. While there are a few examples that show an amount of mathematical precision, there numerous examples where precision is lacking and therefore sloppy conclusions have been made. These math errors, while innocent in intent, must be addressed. At first glance, one would notice that Meisner s website is divided into subcategories such as Design, Theology, Cosmology, and Markets. For my particular quest, the tab titled Design proved to be the most useful. According to an article off the Design tab titled Phi, Pi, and the Great Pyramid of Egypt at Giza (Meisner, 2012), there is evidence to suggest that the Great Pyramid was built with the dimensions that conform to φ. The primary argument is that the Egyptians built the pyramid to have the ratio of the slant height to one half of the base equal to φ, as according to the following figure: (Note: two triangles would be placed back to back to form the face of the pyramid. Therefore, if one half of the base is known and the height is known, through the Pythagorean theorem we can calculate the slant height.) As we examine the trust worthiness of these claims, first we must obtain reasonable measurements of the Great Pyramid. The author of two articles (one opposing the connection of phi to the Great Pyramid and the other for it) both suggest that research concludes the original height is estimated to be 481.4 feet and the length of the base to be 755.79 feet (Markowsky, 1992, p 6 & Meisner, 2012). By use of the Pythagorean theorem, the slant height can be concluded to be approximately 612.01 feet. If we are to find the ratio of the slant height to one half of the base, it is concluded that the ratio is 612.01/ 377.90 1.62 which differs from
φ only by 0.1% (Markowsky, 1992). I was surprised to find that not only did Markowsky s figures NOT refute Meisner s article, but his figures in fact supported it! The question now is not in whether or not the ratio of the slant height to one half of the base is equal to φ, but rather in intention of the builders. Did the creators of the Great Pyramid have the knowledge to be able to conform the pyramid to the golden ratio, or was that simply a coincidental byproduct? Markowsky argues that there is no documentation suggesting that the Egyptians had any knowledge of the golden ratio at this point in time, and therefore certainly would not be constructing a pyramid to φ. This analysis sounds valid, and according to the research of Dr. Dave Lightbody, an archeologist who has done mathematical and archeological work on the Great Pyramid, the fact that phi is involved is merely a byproduct from the real heart of the pyramid design: a deliberate use of circular proportions as was used by Old Kingdom Egyptian architects (Lightbody, 2013). I am not knowledgeable enough of Egyptology to verify any of Dr. Lightbody s claims, however, it does provide an alternative explanation for why phi may be present in the structure and yet not be the focus for its builders. The next claim I would like to examine has to do with the famous work of Leonardo Da Vinci. Meisner in his article claims that the golden section is repeated throughout Da Vinci s painting of The Last Supper (among other paintings from this artist), as is shown in the diagram
below: (picture courtesy of www.goldennumber.net using Meisner s PhiMatrix software) From a first glance, I can understand where Meisner would begin to think of things as being sectioned off in golden sections based off the provided rectangular outlines. However, this process still remains a bit arbitrary. How does one decide that a detail is significant enough to lie near or within a golden rectangle? How exact does a point of the figure need to line up with the golden rectangle in order to claim that it is outlined by a golden rectangle? Does not a thick outline make measurement a bit questionable? It is for these reasons that I among other mathematicians hesitate to claim that Da Vinci s work was based off of the golden section. There are, however, an example or two in which a golden rectangle seems to better approximate the picture, like in the case of Michelangelo s Creation of Adam (picture
retrieved from www.goldennumber.net) How the rectangles relate to the picture is a bit less arbitrary in this example, as we can see the boarders extend from the edge of Adam s wrist, down to his foot, over to God s foot and God s hair. This is, however, provided that the image was not stretched in any way so as to fit the rectangle, nor the dimensions of the shown rectangle differ from a true golden rectangle. Mark Livio in his book The Golden Ratio: The story of Phi, the World s Most Astonishing Number makes a few keen observations about paintings that seem to fit the divine proportion. He cites in his book three paintings that are believed to possess the divine proportion (also known as the golden ratio): Ognissanti, Rucellai and Santa Trinita (Livio, p. 161). However, when of the supposed golden rectangles were measured and their ratios calculated, Livio discovered that the ratios approximated 1.55, 1.59, and 1.73 respectively. Yes, these arbitrary dimensions do near the divine proportion of 1.61803, however, they also near the
simple proportion of 1.6. It is questionable to claim that Da Vinci among other artists in the early Renaissance would have intentionally used the golden ratio. Livio offers that they may instead have used the Vitruvian suggestion for a simple proportion, one that is the ratio of two whole numbers, rather than the Golden Ratio (p. 161). Still, all the same, it seems curious that the ratios would even approximate phi for the rectangular designs used. Could it be that artists were innately drawn to the shape of the golden rectangle because it was pleasing to the eye-- even without a sufficient knowledge or measurement to intentionally use the golden rectangle in design? Is it true that the golden rectangle is the most aesthetically pleasing of all rectangles? Markowsky set out to find the answer to this question by performing an experiment involving a range of rectangle shapes and asking observers to pick out the most pleasing rectangle. In this informal experiment, Markowsky encountered that most participants could not pick out the golden rectangles, and when asked to pick the most pleasing rectangles in both Figure 10 & Figure 11, could not even pick out the same rectangle in both figures. This is an indication that the notion of the golden rectangle being the most
aesthetically pleasing rectangle is unfounded. To the right is a picture found from Markowsky s article (p.14) used to illustrate the experiment. Regardless of mathematical hesitancy, there will always be claims of the connection between phi and other disciplines, as there are already far more claims about the golden ratio in art and architecture than can be covered here. Overall, a word of caution must be presented when examining a claim dealing with φ. There may be some legitimacy to a claim, but if the evidence (or the process of obtaining the evidence) is questionable or arbitrary then the claim should be treated likewise. As far as what I have addressed, the connections between phi, art and architecture in ancient times is considered insufficient and therefore if less importance since we have no way of proving that the ancients were even aware of the golden ratio. Like pi, I anticipate phi will be generating discussion for many more years to come as we discover further mathematical connections. At this point, I will be content in keeping a skeptical yet awed view of one of the most astonishing numbers: φ.
References Livio, M. (2003). The golden ratio: The story of phi, world's most astonishing number. Broadway Books. Lightbody, D. (2013, May 07). The Edinburgh Casing Stone A piece of Giza at the National Museum of Scotland. Retrieved from http://arkysite.wordpress.com/2013/05/07/the-edinburgh-casingstone-a-piece-of-giza-at-the-national-museum-of-scotland/ 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 Meisner, G. (2013, August 5). Phi and the Golden Ratio in Art. Retrieved from http://www.goldennumber.net/art-composition-design/ Meisner, G. (2012, August 18). Phi, pi, and the Great Pyramid of Giza. Retrieved from http://www.goldennumber.net/phi-pi-great-pyramid-egypt/