The Golden Ratio is widely praised for its aesthetic beauty: has its significance been blown out of proportion?
Number enthusiasts are likely familiar with the idea of the golden ratio, and the artistically minded may have heard of its applications in design. The golden ratio is a simple relation between two quantities commonly occurring in mathematics and in nature. For a mathematical ratio, it is remarkably well-known, but its appearance in works for the layperson seem to always mention something about its inherent beauty. Many textbooks, articles and books claim that the golden ratio, and shapes based upon it such as the golden rectangle, are aesthetically pleasing. It has featured in architecture and paintings throughout history, and in human body proportions. These assertions are so widespread that they seem common knowledge, but many of the supposed instances of the golden ratio may be nothing more than myth.
Starting with the basic facts, the golden ratio is defined algebraically with quantities a and b, where a is greater than b, as
In words: two quantities a and b, with a being the greater one, are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The ratio can be represented by the irrational number phi, which is a solution to the quadratic equation x2 – x – 1= 0.
From the golden ratio comes the golden rectangle, whose sides fit the ratio, as well as the golden angle, the golden triangle, pentagram, and pyramid.
The golden ratio was first discovered by the ancient Greeks, in connection with its frequent appearances in geometry. Fibonacci later used the golden ratio to solve geometry problems in the 11th century, although he never related it to the Fibonacci sequence which is named after him. There is a major link between the two, in that the ratios between consecutive Fibonacci numbers (of the sequence 0, 1, 1, 2, 3, 5, 13…) converge to the golden ratio:
In the late 15th century, the golden ratio first gained the pompous name of “the divine proportion” in a book on geometry and architecture, Divina Proportione, written by Italian mathematician and friar Luca Pacioli and illustrated by Leonardo da Vinci. Pacioli gives reasons for the ratio to be referred to as divine, arguing its simplicity, irrationality and self-similarity is representative of aspects of the Christian God. Psychologist Adolf Zeising brought a resurgence in the popularity of the golden ratio in the 19th century, writing of a universal law for “beauty and completeness in the realms of both nature and art”, after noting the golden ratio’s appearance in plants.
The golden ratio and Fibonacci sequence are important in optimization. The spiral arrangements of leaves, seeds and flowers, often given as examples of the Fibonacci sequence in nature, allow the parts to be packed closer together, and so more can fit in a given space.
Some masterpieces of ancient architecture have been associated with the golden ratio, albeit with dubious evidence. The Parthenon in Athens is often featured in introductions to the golden rectangle, with a picture of the façade (in a non-ruined state) enclosed in the rectangle. The Great Pyramid of Giza has also been attributed to golden ratio-incorporating design by the ancient Egyptians, with a slope angle apparently close to one that would make it a golden pyramid. Mathematician George Markowsky wrote a paper in 1992 called Misconceptions about the Golden Ratio, labeling these claims as simply misconceptions. He points out a lack of evidence for the idea that the golden ratio was known of at the time of construction, and that there are many different ways of measuring the Parthenon since it is not rectangular, so the dimensions can effectively be selected to suit the whim of the measurer. The Great Pyramid, he says, is only approximately a golden pyramid, and the only record that this was intentional was written by Herodotus some 2000 years later, in a text which is significantly inaccurate in several aspects of the pyramid’s dimensions.
The golden ratio cannot be applied precisely because it is an irrational number.
Paintings also feature on the classic list of things φ-related. Goldennumber.net contains image after image of famous pieces by Da Vinci, Michelangelo, Mondrian and many others, overlaid with lines to show where the golden proportions come into play. The website was created by Gary Meisner, “The Phi Guy”, the developer of an application to analyse and make use of the golden ratio in design called PhiMatrix. The ratios of the line lengths and spacing marked on the paintings by this software may be exactly the golden ratio, however the choice of their positioning is questionable and could be called arbitrary, an attempt to force a pattern that is not really there. Sometimes lines bisect features; sometimes they align with the edge of an object; sometimes they are placed along the supposed edge of something that the artist has not clearly defined.
Markowsky calls out this ambiguity in his paper, as well as the use of thick lines to draw the superimposed golden rectangle, further reducing accuracy. A commonly claimed example of golden rectangles in Da Vinci’s work is the drawing of an old man with overlain rectangles, which Markowsky says cannot be described with certainty as golden rectangles because they are drawn roughly. Interviewed in 2015, mathematics professor at Stanford University, Keith Devlin, said that although many real-world objects have ratios that “float around it”, the golden ratio cannot be applied precisely because it is an irrational number.
Publications since Adolf Zeising’s 1854 New theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown have asserted that the golden ratio is visually satisfying, the reason the artworks and buildings that incorporate it are considered beautiful. His works also claim that the human body utilises the golden ratio to look attractive: the “optimal” ratio between the total height of the body and the height from the toes to the navel is supposed to be φ. According to Devlin, “when measuring anything as complex as the human body, it’s easy to come up with examples of ratios that are very near to 1.6.”
Possibly concerning are studies of the beauty of faces that contain the golden ratio between various features – on goldennumber.net, the faces sampled appear to be mostly fashion models. Marquardt Beauty Analysis, a company dedicated to the research of human beauty, states that “The ‘Golden Ratio’ is a mathematical ratio of 1.618:1” and that they have used it to construct a mask, a kind of blueprint for a face, which “identifies facial characteristics that are universally perceived as beautiful.” However, the Marquadt mask has been found to have numerous problems, reported in a 2008 paper: “The mask is ill-suited for non-European populations” and “does not appear to describe “ideal” face shape even for white women”, who are its primary targets. Rummaging online for alleged beauty stemming from the golden ratio seems to result in similar examples of narrow standards of beauty. Not only is the human body a victim of golden ratio mania, but other aspects of nature may be as well. A common example given is the nautilus shell, which grows in a logarithmic spiral that makes it self-similar everywhere. Devlin has said in his article The Myth That Will Not Go Away that the constant angle which the spiral turns is not the golden ratio. The Fibonacci numbers do show up in pine cone spirals and often in numbers of flower petals, but there are as many plants that don’t show these patterns as ones that do; the golden ratio is perhaps not as prevalent as sensationalism makes out.
The myth of the golden rectangle being the most beautiful rectangle is addressed by Markowsky as well. The original experiment to determine whether the golden rectangle is superior, conducted by Fechner in the 1860s, asked participants to select their favourite of ten different rectangles, and 76% voted for the three of medium proportions – hardly compelling evidence, says Markowsky, who challenges the reader to pick which they think is the golden rectangle out of 48 options. It is near impossible to do by eye. Multiple studies since have not shown much evidence of preference for the golden ratio in rectangles. Ironically, the popularity of claims about the golden ratio in association with mysticism has led to it being intentionally and explicitly included in some more modern works of art and architecture. Salvador Dalí’s surreal painting The Sacrament of the Last Supper is on a canvas of golden rectangle dimensions and features a giant dodecahedron – another instance of the mathematical linked with the mystical – which is in perspective so that its edges are in the golden ratio with each other.
The golden ratio, in the eyes of non-mathematicians, seems to have risen to the status of a mysterious entity whose supposed presence signifies something special. Despite in reality being simply a useful and interesting ratio in geometry and other branches of mathematics, it has been touched by fringe theories and sensationalism since the Victorian era. The many purported cases of the golden ratio are down to people’s “natural desire to find meaning in the pattern of the universe”, Devlin says. Whether the golden rectangle is beautiful and the 1.618 ratios in paintings by Da Vinci and others are intentional, or whether people just see what they want to see, these things would seem to be more down to personal opinion than science.
Written by Catriona Roy and edited by Ailie McWhinnie.
Catriona’s thoughts… I believed the misconceptions about the golden ratio that I have highlighted here for a long time, after reading them in an old popular science book. Researching the topic for this article, I was dismayed to learn that these “fun facts” are more like “frustrating falsehoods”, so I decided to focus my piece on separating truth from myth. However, even after all this, I don’t think the golden ratio is any less beautiful from a mathematical perspective. Unlike in ancient art and history, the many patterns and great interwovenness in maths can be indisputably proven to exist, which I think gives maths an allure at least equalling that of earthly applications like botany and painting. In any case, fact is preferable to fiction, which is what science is all about.