Over the next half hour, Langdon showed them slides of artwork by Michelangelo, Albrecht Dürer, Da Vinci, and many others, demonstrating each artist’s intentional and rigorous adherence to the Divine Proportion in the layout of his compositions. Langdon unveiled PHI in the architectural dimensions of the Greek Parthenon, the pyramids of Egypt, and even the United Nations Building in New York. PHI appeared in the organizational structures of Mozart’s sonatas, Beethoven’s Fifth Symphony, as well as the works of Bartok, Debussy, and Schubert. The number PHI, Langdon told them, was even used by Stradivarius to calculate the exact placement of the f-holes in the construction of his famous violins (Brown, 2003, p. 101).
The Da Vinci Code, Dan Brown’s 2003 best-selling novel, features the Golden Ratio as a key plot component. This novel and its 2006 movie adaptation helped bring awareness of this important mathematical and artistic proportion into the public sphere. Indeed, many people who had not encountered the Golden Ratio during their school years were introduced to it courtesy of Brown’s work. While Brown should be commended for bringing mathematics to the masses, there are several problems with his treatment of the Divine Proportion. First, he defines PHI simply as 1.618 without reference to its irrationality or origins. Second, he makes many questionable claims about the prevalence of the Golden Ratio in art, architecture, and nature.
To be fair, Brown is a novelist, not a mathematician or an educator, and The Da Vinci Code is quite clearly a work of fiction. One should not depend on it for mathematical precision and detail. As well, he is not the only person ever to have published flawed information about the Golden Ratio. However, unlike others who have publicized dubious mathematical claims, he has been able to impact an unprecedented number of people through his novel and movie. Many people, including no small number of educators, have received his message and have, in some cases, constructed misguided and incomplete mathematical understandings as a result. It is, therefore, fitting that a more genuine description of the origin of the Golden Ratio as well as a few practical applications be presented.
To be fair, Brown is a novelist, not a mathematician or an educator, and The Da Vinci Code is quite clearly a work of fiction. One should not depend on it for mathematical precision and detail. As well, he is not the only person ever to have published flawed information about the Golden Ratio. However, unlike others who have publicized dubious mathematical claims, he has been able to impact an unprecedented number of people through his novel and movie. Many people, including no small number of educators, have received his message and have, in some cases, constructed misguided and incomplete mathematical understandings as a result. It is, therefore, fitting that a more genuine description of the origin of the Golden Ratio as well as a few practical applications be presented.
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Although Brown’s novel did much to bring the Golden Ratio, and the controversy that often accompanies it, into the public consciousness, disagreement over the prevalence of this proportion in nature and art has pitted academics against each other for decades. Some see the Divine Proportion represented in many human and natural endeavours, while others insist that most such occurrences are flawed and that the forced attempt to find the Golden Ratio where it does not actually exist only serves to diminish its power and beauty. Below, we examine some of the major points of contention between these two sides.
The Nautilus
In Chapter 20 of The Da Vinci Code, Robert Langdon recalls a lesson in which he discusses manifestations of the Golden Ratio in nature and art. One of his first claims is that the spiral of the nautilus (a cephalopod mollusk) adheres to the Divine Proportion (Brown, 2003). Bentley (2016) agrees with this statement and even presents an image of a nautilus superimposed on a Golden Spiral (seen to the right) to illustrate his point. A quick glance, however, reveals that the spirals are not the same; the Golden Spiral widens at a greater rate as it expands outward from its centre. George Hart (2012) addresses this fallacy head on by using a 3D printer to create a version of the nautilus that does correspond to the Golden Ratio. He then compares it to a real nautilus and describes the mathematical differences between them. There is little room for interpretation or opinion in the case of the nautilus. It is quite clear that all claims of its adherence to the Golden Ratio are simply incorrect.
In Chapter 20 of The Da Vinci Code, Robert Langdon recalls a lesson in which he discusses manifestations of the Golden Ratio in nature and art. One of his first claims is that the spiral of the nautilus (a cephalopod mollusk) adheres to the Divine Proportion (Brown, 2003). Bentley (2016) agrees with this statement and even presents an image of a nautilus superimposed on a Golden Spiral (seen to the right) to illustrate his point. A quick glance, however, reveals that the spirals are not the same; the Golden Spiral widens at a greater rate as it expands outward from its centre. George Hart (2012) addresses this fallacy head on by using a 3D printer to create a version of the nautilus that does correspond to the Golden Ratio. He then compares it to a real nautilus and describes the mathematical differences between them. There is little room for interpretation or opinion in the case of the nautilus. It is quite clear that all claims of its adherence to the Golden Ratio are simply incorrect.
The Parthenon
Several scholars agree with Dan Brown’s contention (presented in the quotation at the top of this webpage) that the Parthenon in Athens was built according to the Divine Proportion (Bentley, 2016; Hyde, 2004; Jarvis, 2007). George Markowsky (1992), in his oft-cited article, Misconceptions about the Golden Ratio, offers a firm rebuttal. He describes other scholars’ attempts to capture the Parthenon inside a Golden Rectangle and wonders why none of them seems to be bothered by the fact that parts of the building actually do not fit inside the quadrilaterals they have constructed. Additionally, he says that there is a multitude of measurements of the Parthenon resulting from “different authors…measuring from different points” (p. 8). With so many available figures, he claims, it is easy for the “golden ratio enthusiast” to make the Divine Proportion fit the Parthenon instead of the other way around (p. 9). Despite his stated lack of certainty about the dimensions of the structure, Markowsky selects figures for height, length, and width and determines that the ratios constructed with them fall outside of his acceptance range of 1.58 to 1.66.
In 2016, Gary Meisner, creator of goldennumber.net, wrote a response to Markowsky’s article. He reveals some flaws in Markowsky’s argument concerning the Parthenon. He notes that the single image of the Parthenon presented in the 1992 article has been “stretched horizontally to deform it to a ratio of 1.77, and does not represent the Parthenon’s real proportions” (Meisner, 2016). The pictures below show how Markowsky’s image differs from an actual photograph of the Parthenon (as well as how a Golden Rectangle appears to fit quite nicely in the photograph).
Several scholars agree with Dan Brown’s contention (presented in the quotation at the top of this webpage) that the Parthenon in Athens was built according to the Divine Proportion (Bentley, 2016; Hyde, 2004; Jarvis, 2007). George Markowsky (1992), in his oft-cited article, Misconceptions about the Golden Ratio, offers a firm rebuttal. He describes other scholars’ attempts to capture the Parthenon inside a Golden Rectangle and wonders why none of them seems to be bothered by the fact that parts of the building actually do not fit inside the quadrilaterals they have constructed. Additionally, he says that there is a multitude of measurements of the Parthenon resulting from “different authors…measuring from different points” (p. 8). With so many available figures, he claims, it is easy for the “golden ratio enthusiast” to make the Divine Proportion fit the Parthenon instead of the other way around (p. 9). Despite his stated lack of certainty about the dimensions of the structure, Markowsky selects figures for height, length, and width and determines that the ratios constructed with them fall outside of his acceptance range of 1.58 to 1.66.
In 2016, Gary Meisner, creator of goldennumber.net, wrote a response to Markowsky’s article. He reveals some flaws in Markowsky’s argument concerning the Parthenon. He notes that the single image of the Parthenon presented in the 1992 article has been “stretched horizontally to deform it to a ratio of 1.77, and does not represent the Parthenon’s real proportions” (Meisner, 2016). The pictures below show how Markowsky’s image differs from an actual photograph of the Parthenon (as well as how a Golden Rectangle appears to fit quite nicely in the photograph).
Meisner then addresses the three ratios Markowsky created (seen to the right). The first two, he claims, were never intended to adhere to the Golden Ratio, and so it is no surprise that they fall well outside of the acceptance range. The ratio of height to width is closer to the Divine Proportion, but is still too far away from it for Markowsky’s liking. However, Meisner identifies an issue with the measurements that Markowsky used. He says that Markowsky’s figure for height was taken from the base of the columns whereas the height from the base of the steps is a more accurate measure of the Parthenon’s dimensions. Using this more correct figure brings the ratio into Markowsky’s acceptance range. Meisner’s final main argument against Markowsky’s thoughts on the Parthenon is that the Golden Ratio is present in many smaller aspects of the structure, and that Markowsky only addressed ratios created from the three largest measurements. According to Meisner, Markowsky ignored some of the richest and most stunning manifestations of the Divine Proportion that the Parthenon has to offer.
The debate about the Parthenon’s adherence to the Golden Ratio persists to this day. Enthusiasts from both sides continue to develop and defend new arguments. Unlike the nautilus, the case for the Divine Proportionality of the Parthenon is likely to remain open for some time.
The debate about the Parthenon’s adherence to the Golden Ratio persists to this day. Enthusiasts from both sides continue to develop and defend new arguments. Unlike the nautilus, the case for the Divine Proportionality of the Parthenon is likely to remain open for some time.
Painters
Markowsky (1992) takes issue with claims that the Golden Ratio has often been purposefully used in the creation of art. He takes particular aim at Leonardo da Vinci. In response to the popular notion that da Vinci incorporated the Divine Proportion into his works, Markowsky submits counter-arguments focusing on two pieces. The first of these is a self-portrait drawing in profile. Markowsky says that the rectangles drawn on it are crudely accomplished (some feature rounded corners) and, as such, cannot be referenced for any consideration of the Golden Ratio.
Markowsky (1992) takes issue with claims that the Golden Ratio has often been purposefully used in the creation of art. He takes particular aim at Leonardo da Vinci. In response to the popular notion that da Vinci incorporated the Divine Proportion into his works, Markowsky submits counter-arguments focusing on two pieces. The first of these is a self-portrait drawing in profile. Markowsky says that the rectangles drawn on it are crudely accomplished (some feature rounded corners) and, as such, cannot be referenced for any consideration of the Golden Ratio.
The second work in question is da Vinci’s unfinished canvas of St. Jerome. Markowsky takes exception to the enclosure of St. Jerome within a Golden Rectangle. He sees the placement of the rectangle as “arbitrary” (p. 11). The images below reveal his reasons. The saint’s head does not touch the top of the rectangle, and most of his right arm is completely outside of it. To Markowsky, this is proof of his point that many “authors will draw golden rectangles that conveniently ignore parts of the object under consideration” (p. 5). George Hart (2015) concurs with this view. In his video linked here, he claims that the vague natures of many artistic and natural objects allow the Golden Ratio to be found if one looks long enough. As he narrates this idea, the video shows Golden Rectangles being placed in random locations on the Mona Lisa. He also bemoans the tendency for people to assume the Golden Ratio whenever any measurement approaches one and two-thirds. Hart’s points are well-taken. In many cases, the location of the Golden Ratio within a work of art is more a construction of the viewer than the artist.
Meisner (2016), takes issue with the fact that Markowsky uses only two works of art (both of which are incomplete) to create a sweeping generalization about da Vinci’s entire body of work. He does not attempt to defend either of the two works under examination; in fact, he refers to them as “feable strawman images” (Meisner, 2016). However, he disagrees with the notion that da Vinci was largely ignorant of the Golden Ratio and that none of his works contain it. A few simple measurements of da Vinci’s most famous (especially religious) paintings, he says, would be enough to show the great extent to which he purposely embedded the Divine Proportion into his work. In the video to the right, Meisner uses his PhiMatrix software to demonstrate the many manifestations of the Golden Ratio in da Vinci’s Salvator Mundi.
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It is difficult to know exactly what da Vinci intended vis-à-vis the Golden Ratio because we do not have access to his thoughts on the matter. We are left with scattered facts that are used by scholars to support their own opinions, but the truth ultimately remains a mystery. The case of George Seurat is an interesting one. He created more than two dozen works on wooden rectangles featuring golden proportions (Meisner, 2016). Nonetheless, Markowsky (1992), quoting Roger Fischler, notes that there is no evidence of the use of the Golden Ratio as the basis of any of Seurat’s paintings. Without the artist’s personal thoughts on the matter, we are left to choose the side that makes the most convincing argument. However, artists in more recent times have left more complete records. Alex Colville (1920-2013) is known for having used the Golden Ratio in his works. Todkill (2000) notes that Colville orders space through a variety of “arcane means,” one of which is the “‘golden section’ of classical art” (p. 748). His inclusion of the Divine Proportion into his art cannot be denied as he left the following description of the aesthetic experience that accompanied its use:
"Once you begin to perceive these relationships, circles, spirals, triangles, and rectangles appear as if on their own. The beauty of it comes as a surprise, and its harmonies inspire joy. Part of the excitement is that these discoveries, though new to us, are about immutable laws that have been in force since space and time began" (Fry, 1994, quoted in Jarvis & Naested, 2012, p. 14).
"Once you begin to perceive these relationships, circles, spirals, triangles, and rectangles appear as if on their own. The beauty of it comes as a surprise, and its harmonies inspire joy. Part of the excitement is that these discoveries, though new to us, are about immutable laws that have been in force since space and time began" (Fry, 1994, quoted in Jarvis & Naested, 2012, p. 14).
The Most Aesthetically Pleasing Rectangle
Many people are familiar with the idea that the Golden Rectangle is more aesthetically pleasing than other rectangles. Markowsky (1992) traces the origin of this belief to an 1860s experiment in which subjects were asked to indicate a preference for one rectangle out of ten. The ratios of these shapes ranged from 1.00 (perfect square) to 0.40. 76% of the subjects chose one the three that were closest to the Golden Ratio (0.57, 0.62, 0.68). Markowsky repeated this experiment with a greater number and variety of rectangles. Very few people selected Golden Rectangles from either of the two presentations of the 48 figures, but a high concentration of people seemed to prefer a specific range of rectangles (which happened to include the Golden Rectangle between its extremes). He interprets this result as evidence against the aesthetic appeal of the Golden Rectangle. Meisner (2016) views the whole endeavour as pointless and misguided. Asking people to choose a favourite rectangle from a group, he claims, is akin to playing a selection of random musical notes and requiring the listener to indicate the one he/she prefers. Most people simply do not have strong feelings about rectangles (or musical notes) presented in the absolute absence of context and/or purpose. He suggests using a context to which the Golden Rectangle can be applied. The human head is a good one because its height and width often feature near golden proportions. Subjects looking at the picture below could choose the most aesthetically pleasing image and then superimpose Golden Rectangles on them in order to see the inherent beauty of the Divine Proportion (the rectangles of the heads below have the following ratios: 2.59, 2.30, 2.00, 1.80, 1.62, 1.40, 1.26, 1.11).
Many people are familiar with the idea that the Golden Rectangle is more aesthetically pleasing than other rectangles. Markowsky (1992) traces the origin of this belief to an 1860s experiment in which subjects were asked to indicate a preference for one rectangle out of ten. The ratios of these shapes ranged from 1.00 (perfect square) to 0.40. 76% of the subjects chose one the three that were closest to the Golden Ratio (0.57, 0.62, 0.68). Markowsky repeated this experiment with a greater number and variety of rectangles. Very few people selected Golden Rectangles from either of the two presentations of the 48 figures, but a high concentration of people seemed to prefer a specific range of rectangles (which happened to include the Golden Rectangle between its extremes). He interprets this result as evidence against the aesthetic appeal of the Golden Rectangle. Meisner (2016) views the whole endeavour as pointless and misguided. Asking people to choose a favourite rectangle from a group, he claims, is akin to playing a selection of random musical notes and requiring the listener to indicate the one he/she prefers. Most people simply do not have strong feelings about rectangles (or musical notes) presented in the absolute absence of context and/or purpose. He suggests using a context to which the Golden Rectangle can be applied. The human head is a good one because its height and width often feature near golden proportions. Subjects looking at the picture below could choose the most aesthetically pleasing image and then superimpose Golden Rectangles on them in order to see the inherent beauty of the Divine Proportion (the rectangles of the heads below have the following ratios: 2.59, 2.30, 2.00, 1.80, 1.62, 1.40, 1.26, 1.11).
The Human Body
During his Harvard University lecture in Chapter 20 of The Da Vinci Code, Robert Langdon alludes to the notion of the Divine Proportion being part of the construction of the human body. Specifically, he notes that the division of one’s height by the distance between one’s navel and the floor yields the Golden Ratio (Brown, 2003). Markowsky (1992) rejects this contention, but he offers weak evidence to support his opinion. His trials on members of his immediate family yield ratios of 1.59, 1.63, 1.65, and 1.66. These figures, as well as their average of 1.63, fall well within his acceptance range. However, he dismisses these results due to the small nature of the sample and his perception that the human navel has an imprecise location. Meisner (2016) argues that the centre of the navel could easily function as a specific location for this calculation. Also, in the video below, he uses his PhiMatrix software to analyze the human face for manifestations of the Golden Ratio. The results are quite shocking and lend credence to the notion of the Divine Proportion as a building block of the human form.
During his Harvard University lecture in Chapter 20 of The Da Vinci Code, Robert Langdon alludes to the notion of the Divine Proportion being part of the construction of the human body. Specifically, he notes that the division of one’s height by the distance between one’s navel and the floor yields the Golden Ratio (Brown, 2003). Markowsky (1992) rejects this contention, but he offers weak evidence to support his opinion. His trials on members of his immediate family yield ratios of 1.59, 1.63, 1.65, and 1.66. These figures, as well as their average of 1.63, fall well within his acceptance range. However, he dismisses these results due to the small nature of the sample and his perception that the human navel has an imprecise location. Meisner (2016) argues that the centre of the navel could easily function as a specific location for this calculation. Also, in the video below, he uses his PhiMatrix software to analyze the human face for manifestations of the Golden Ratio. The results are quite shocking and lend credence to the notion of the Divine Proportion as a building block of the human form.
The points discussed above represent only a fraction of the Golden Ratio issues over which people continue to disagree. There are many others that are not represented here. In only a few cases, such as that of the nautilus, are we able to obtain definitive answers. The others will continue to spark debate well into the future. For the purposes of the mathematics classroom, this is actually a very good thing. So long as confusion and disagreement reign in this realm of mathematical thought, students and teachers have plenty of material for investigation. Instead of teaching students that the Divine Proportion does or does not correspond to the Parthenon, da Vinci’s art, the human body, or any of the other contexts under debate, teachers will serve their students much better by educating them about the nature of the disagreement in a given context and allowing the students to investigate it to determine their own thoughts on the matter. Whether the students agree with the teacher or each other is immaterial. What counts is that the act of investigation allows them to encounter, appreciate, and develop an understanding of an important mathematical idea. It is in this spirit that we now move into a plan for classroom investigation on the next page.
References
Bentley, B. (2016). Why the golden proportion really is golden. Australian Mathematics Teacher, 72(1), 10-14. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1826529449?accountid=15115
Brown, D. (2003). The Da Vinci code. New York: Random House.
Hart, G. (2012, July 22). The golden ratio nautilus. [Video File]. Retrieved from https://www.youtube.com/watch?v=_gxC8OjoQkQ
Hart, G. (2015, March 27). Mathematical impressions: The golden ratio. [Video File]. Retrieved from https://www.simonsfoundation.org/multimedia/mathematical-impressions-multimedia/mathematical-impressions-the-golden-ratio/
Hyde, H. (2004). The golden ratio. Australian Mathematics Teacher, 60(1), 30-31. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/62068744?accountid=15115
Jarvis, D. (2007). Mathematics and visual arts: Exploring the golden ratio. Mathematics Teaching in the Middle School, 12(8), 467-473. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/231093806?accountid=15115
Jarvis, D., & Naested, I. M. (2012). Exploring the math and art connection: teaching and learning between the lines. Calgary, AB: Detselig.
Todkill, A. M. (2000). The existential art of Alex Colville. Canadian Medical Association Journal, 163(6), 748. Retrieved from http://www.cmaj.ca/content/163/6/748.full.pdf
Markowsky, G. (1992). Misconceptions about the golden ratio. College Mathematics Journal, 23(1), 2-19. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/62946073?accountid=15115
Meisner, G. (2016). Dr. George Markowsky’s “Misconceptions about the Golden Ratio” Reviewed. Retrieved from https://www.goldennumber.net/golden-ratio-misconceptions-by-george-markowsky-reviewed/
Bentley, B. (2016). Why the golden proportion really is golden. Australian Mathematics Teacher, 72(1), 10-14. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1826529449?accountid=15115
Brown, D. (2003). The Da Vinci code. New York: Random House.
Hart, G. (2012, July 22). The golden ratio nautilus. [Video File]. Retrieved from https://www.youtube.com/watch?v=_gxC8OjoQkQ
Hart, G. (2015, March 27). Mathematical impressions: The golden ratio. [Video File]. Retrieved from https://www.simonsfoundation.org/multimedia/mathematical-impressions-multimedia/mathematical-impressions-the-golden-ratio/
Hyde, H. (2004). The golden ratio. Australian Mathematics Teacher, 60(1), 30-31. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/62068744?accountid=15115
Jarvis, D. (2007). Mathematics and visual arts: Exploring the golden ratio. Mathematics Teaching in the Middle School, 12(8), 467-473. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/231093806?accountid=15115
Jarvis, D., & Naested, I. M. (2012). Exploring the math and art connection: teaching and learning between the lines. Calgary, AB: Detselig.
Todkill, A. M. (2000). The existential art of Alex Colville. Canadian Medical Association Journal, 163(6), 748. Retrieved from http://www.cmaj.ca/content/163/6/748.full.pdf
Markowsky, G. (1992). Misconceptions about the golden ratio. College Mathematics Journal, 23(1), 2-19. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/62946073?accountid=15115
Meisner, G. (2016). Dr. George Markowsky’s “Misconceptions about the Golden Ratio” Reviewed. Retrieved from https://www.goldennumber.net/golden-ratio-misconceptions-by-george-markowsky-reviewed/