Were the Greeks Really Obsessed with the Golden Ratio?

The so-called “Golden Ratio,” or φ, occurs when the ratio of the greater of two quantities to the lesser of two quantities is equivalent to the ratio of the sum of the two quantities to the greater of the two quantities. Expressed using incomprehensible math symbols, it looks like this:

Many people believe that the “Golden Ratio” is the pinnacle of aesthetic perfection and that, the closer something is to the Golden Ratio, the more beautiful it is automatically. Many people also believe that the ancient Greeks were obsessed with the Golden Ratio and that they incorporated it into all their buildings and works of art. Unfortunately for those who love a good math story, we have no good evidence to support either of these conclusions.

In fact, the Golden Ratio is not even mentioned in any Greek text until as late as the early third century BC. The Greeks were arguably fascinated with the idea of using mathematical proportions in art to a certain extent, but they were by no means obsessed with the Golden Ratio in particular. The story of how we came to believe that the Greeks were obsessed with the Golden Ratio, though, is as fascinating as it is bizarre. It involves a friend of Leonardo da Vinci, an eccentric nineteenth-century German psychologist, and Donald Duck.

Drawing rectangles and spirals is fun!

You often see images on the internet like the ones shown below, which supposedly illustrate how the design of the Parthenon perfectly fits the Golden Ratio. I found the images below just by doing a Google Images search for “Parthenon Golden Ratio”:

Here is a close-up of the first two images I found:

The problem is that the spirals and rectangles in these images don’t actually line up with the building, which you can tell just by looking at the pictures. If the rectangle lines up in one place, it doesn’t line up in another. Often, these images deliberately use very thick lines or are taken from a particular angle in order to disguise this fact.

The thing is, drawing a bunch of rectangles and spirals on photographs of buildings and works of art doesn’t automatically prove that those buildings were constructed according to the Golden Ratio. You can superimpose rectangles and spirals over literally anything. For instance, here is an image of a Golden Ratio spiral superimposed over a dumpster:

Can you even begin to fathom how incredibly aesthetically appealing this dumpster is? Is it not the most beautiful dumpster you have ever seen?

Here is an image of a Golden Ratio spiral superimposed over the illustration of Satan from the Codex Gigas:

Isn’t Satan so amazingly beautiful? Who could ever hope to resist such a magnificent Devil!

Here is an image of a Golden Ratio spiral superimposed over my own face:

Clearly, this proves beyond a shadow of a doubt that my face is designed according to the Golden Ratio and is mathematically perfect. This image clearly proves that I have the most aesthetically appealing face in the world. All hail my magnificent face!

Ok. Enough silliness. I think you get my point; imposing a Golden Ratio rectangle or spiral over an image of an object doesn’t automatically make the object in that image correspond to the Golden Ratio and, if you look closely at the images of the Parthenon with the Golden Ratio rectangles and spirals superimposed on them, you will see that the rectangles and spirals do not really fit the building.

The Parthenon—not built according to the Golden Ratio

Of course, looks can be deceiving. What we really need here is something better than just a photograph of the building with geometric figures superimposed over it; what we really need are the exact measurements of the Parthenon and the mathematical relationships between them.

I would give an in-depth mathematical debunking of the idea that the Golden Ratio was used in the construction of the Parthenon, using actual measurements from the Parthenon rather than just photos of it. Unfortunately, I was never a particularly talented math student and I doubt I could do a mathematical debunking convincingly, so, instead, I will quote for you a very sound debunking from this very well-written article by New Zealand classicist Peter Gainsford, which looks at the actual measurements of the Parthenon and demonstrates that it is not constructed according to the Golden Ratio. Gainsford writes:

“Myth 1(a): In the Parthenon frieze, each square metope + rectangular triglyph together form a golden rectangle. The triglyph is another golden rectangle.”

“Reality: To avoid problems with foreshortening, let’s get some accurate measurements. I’m taking my figures from Lehman and Weinman 2018: 167.”

“On the west facade, the average metope width is 1275 mm, and the average triglyph width is 844.6 mm, making a total rectangle of 1275 × 2119.6 mm. A golden rectangle of the same height ought to be 1275 × 2063 mm, or if the same width, 1310 × 2119.6 mm. On the east facade, the figures are almost the same: average metope width 1274 mm, average triglyph width 844.5 mm, total rectangle 1274 × 2118.5 mm. The triglyphs are more than 7% too fat to be golden rectangles.”

“The actual ratio intended between metope and triglyph is 3:2. On the west facade it’s 3.019:2, on the east facade 3.017:2. Combined, each metope + triglyph would then produce a 5 × 3 rectangle, not φ × 1. They miss φ by 2.7%, but they miss 5 × 3 by only 0.23% to 0.25%.”

“Myth 1(b): A rectangle the width of a metope + triglyph, and the height of the entablature, is a golden rectangle.”

“Reality: The height of the entablature is 3295 mm, so based on the figures above, the rectangle is 2119.6 × 3295 mm (west facade) or 2118.5 × 3295 mm (east). A golden rectangle of the same height ought to be 2036 mm wide, or if the same width, 3430 mm high (west) or 3428 mm high (east). The entablature is 4% too short, or alternatively, the metopes + triglyphs are 4% too wide.”

“Myth 1(c): Each pair of columns and the space between them form a golden rectangle.”

“Reality: The columns are 10.433 m tall. The diameter at the bottom is 1.905 m, and the average intercolumniation is 4.296 m (not counting the corner columns, which are more narrowly spaced). This gives a rectangle of 6.201 m × 10.433 m. A golden rectangle with that width ought to be 6.201 × 10.033 m (so the real columns are 4% too tall), or with that height, 6.448 m × 10.433 m (so the real columns are 4% too close together).”

In addition to pointing out how the measurements of the Parthenon do not fit the Golden Ratio, Gainsford also observes that the architects who created the Parthenon could have made the Parthenon fit the Golden Ratio if they had wanted to:

“If the architects of the Parthenon had wanted to embed the golden ratio in the building, they certainly could have done so: ancient Greek temples do display various other ratios, to fairly high precision, as documented by Lehman and Weinman (2018: 61-104). But they’re ratios like 2:1, 9:4, 7:3, and in some parts of the Parthenon, 81:30. The golden ratio doesn’t enter into it.”

In other words, the reason why the Parthenon does not fit the Golden Ratio is clearly because the architects who designed it were not trying to make it fit the Golden Ratio.

A word from a mathematician

Classicists are not the only ones who agree that the Parthenon is not built according to the Golden Ratio; mathematicians also agree. In a blog post from May 2007, Keith Devlin, a professor of mathematics at Stanford University, writes concerning the notion that the Greeks constructed the Parthenon according to the Golden Ratio: “…they did not; which is to say, there is not a shred of evidence that they did any such thing, and good reason to believe they did not.”

Devlin goes on to comment later in the article regarding the images that are so pervasive on the internet depicting the Parthenon with rectangles and spirals superimposed on it. He writes:

“As to the Parthenon, all it takes is more than a cursory glance at all the photos on the Web that purport to show the golden ratio in the structure, to see that they do nothing of the kind. (Look carefully at where and how the superimposed rectangle – usually red or yellow – is drawn and ask yourself: why put it exactly there and why make the lines so thick?)”

In other words, Devlin is saying basically the same thing I just said: that drawing rectangles on an image doesn’t prove the building or object in the image is constructed according to the Golden Ratio.

The beginning of the myth of the Golden Ratio

As far as I am aware, there is actually no evidence that the Greeks even knew about the Golden Ratio at the time when the Parthenon was constructed at all. The earliest known description of the Golden Ratio comes from the Greek mathematical writer Eukleides of Alexandria (lived c. 325 – c. 270 BC) in Book Six of his mathematics textbook Elements of Geometry. Eukleides doesn’t call it “the Golden Ratio”; instead he calls it “the extreme and mean ratio.” Also, Eukleides says nothing about the ratio being widely used in art or even having any aesthetic relevance at all.

The Golden Ratio first began to be mythologized by the Italian Renaissance mathematician Luca Pacioli (lived c. 1447 – 1517), a friend of Leonardo da Vinci. In 1509, Pacioli published a treatise titled De Divina Proportione, which is Latin for “On the Divine Proportion.” The title referred to none other than the Golden Ratio and the book was illustrated by none other than Leonardo da Vinci himself.

ABOVE: Portrait from 1495 of the Italian Renaissance mathematician Luca Pacioli, whose treatise De Divina Proportione arguably marks the beginning of the modern mythologization of the Golden Ratio

An eccentric German

The real beginning of our modern obsession with the Golden Ratio, though, came with the eccentric German psychologist Adolf Zeising (lived 1810 – 1876), who became thoroughly convinced that the Golden Ratio was everywhere in nature. He thought that the Golden Ratio could be found in nearly everything living, from plant leaves to the human skeleton.

In 1854, Zeising published a book in German titled Neue Lehre von den Proportionen des menschlichen Körpers (i.e. “New Teachings on the Proportions of the Human Body”) in which he argued that the Golden Ratio is “das Grundprinzip aller nach Schönheit und Totalität drängenden Gestaltung im Reich der Natur, wie im Gebiet der Kunst enthalten ist,” which means “the founding principle for all creative striving for beauty and wholeness, both in the realm of nature as well as in the domain of art.”

Zeising wrote about the Golden Ratio in subsequent work. His own obsession with the Golden Ratio gradually spread to other mathematicians and art theorists, eventually giving birth to the myth of the Golden Ratio as we know it today. Art historians went back through old works of art and architecture and retrospectively concluded that many of them were constructed according to the Golden Ratio, even though they really weren’t. Thus, just about any rectangle or spiral in an old work of art was automatically interpreted as a Golden Ratio.

ABOVE: Illustration from 1866 showing the German writer Adolf Zeising at a party. Zeising was the one who popularized the idea that the Golden Ratio is the basis for all aesthetics.

Pheidias and φ

In around 1910, the engineer Mark Barr (lived 1871 – 1950), a dual citizen of the United States and Great Britain, proposed the idea of representing the Golden Ratio with the Greek letter phi (φ). Barr reportedly proposed this partly because φ is the first letter of the name of the great Athenian sculptor Pheidias (lived c. 480 – c. 430 BC), whose name in Greek is Φειδίας (Pheidías). Pheidias was in charge of designing and creating the sculptures for the Parthenon.

Barr seems to have therefore been operating under the assumption that Pheidias used the Golden Ratio when designing the sculptures for the Parthenon. There is no evidence to support this assumption and it is quite likely that Pheidias did not know of the Golden Ratio’s existence, since, as I have already noted, the earliest known description of it comes from Eukleides’s Elements of Geometry, which was written over a century after Pheidias’s death.

Nonetheless, Barr’s notation eventually became accepted and it is now standard in mathematics to use the Greek letter φ as a symbol for the Golden Ratio. Thus, the Golden Ratio’s symbol comes from the first letter of the name of a man who may very well have not even known of its existence.

ABOVE: Pheidias Showing the Frieze of the Parthenon to his Friends, painted in 1868 by the English Academic painter Sir Lawrence Alma-Tadema, correctly depicting the Parthenon Frieze as having originally been painted

Donald Duck joins in on the fun!

The myth of the Golden Ratio was further propagated through the twenty-seven-minute-long animated children’s educational film Donald in Mathmagic Land, which was produced by Walt Disney in 1959. This film claimed that the ancient Greeks regarded the Golden Rectangle as “a mathematical law of beauty.” In reality, it was not the Greeks, but rather Adolf Zeising who believed this.

The film goes on to claim that many famous buildings and works of art, including the Parthenon, the Notre Dame de Paris cathedral, the Mona Lisa, and the Venus de Milo all contain Golden Ratios. In reality, there is no evidence that any of these buildings or works of art really contain Golden Ratios. Nevertheless, Donald in Mathmagic Land proceeds to draw rectangles all over them, claiming that they are built according to the Golden Ratio. Because the versions of the buildings and works of art shown in the segment are animated, the film is able to make the rectangles look somewhat more convincing than they would if they were drawn over a real photograph.

Nonetheless, despite its egregious errors, Donald in Mathmagic Land has been commonly shown to schoolchildren throughout the English-speaking world ever since it first came out roughly three generations ago. People tend to have a strong attachment to information they learned as children, so many people have a very strong attachment to the idea that buildings such as the Parthenon were constructed according to the Golden Ratio, even though there is no evidence to support this.

ABOVE: Image of Donald Duck from the Golden Ratio segment of Donald in Mathmagic Land, which propagated the misconception that many ancient Greek buildings and sculptures were deliberately designed according to the Golden Ratio

Not a magic number

Unfortunately, most of what you hear about the Golden Ratio is pure malarkey. The Golden Ratio isn’t a magical, mystical, esoteric number that all of nature intentionally patterns itself after; that’s not how nature works. The Golden Ratio does show up in nature sometimes because Fibonacci Sequences occur in nature and Fibonacci Sequences tend to result in the Golden Ratio, but it doesn’t show up nearly as often as Zeising and others obsessed with the Golden Ratio have claimed.

Furthermore, there isn’t really good scientific evidence that the Golden Ratio is naturally aesthetically appealing to humans. In fact, there is actually some experimental evidence to suggest that people have no natural preference for the Golden Ratio at all. For instance, Keith Devlin, the same mathematics professor at Stanford University whom I quoted earlier, has conducted an exercise with students at his university, in cooperation with the university’s psychology department, for years.

As part of the exercise, students are shown images of different rectangles. Only one of the rectangles is constructed according to the Golden Ratio. Each student is instructed to pick their favorite rectangle of all the ones shown. Devlin has found that students are about as likely to pick the Golden Rectangle as any of the others, which suggests that the Golden Ratio is not inherently as aesthetically appealing as early theorists believed.

Greek use of numbers in art

Now that we have seen where the idea that the Greeks were obsessed with the Golden Ratio comes from, let’s get back to the Greeks. It is true that at least some ancient Greek artists tried to use numbers and mathematics in their art. For instance, the fifth-century BC Greek sculptor Polykleitos of Sikyon is known to have written a treatise titled Canon of Proportion in which he laid out in exact detail the ideal proportions for a realistic-looking but idealized sculpture of a nude male. Polykleitos is known to have used these proportions in his sculptures.

Polykleitos’s Canon of Proportions has not survived to the present day, but we know about it because it is mentioned and even occasionally quoted in a number of surviving ancient texts. Unfortunately for people who want to believe the Greeks were obsessed with the Golden Ratio, though, no surviving ancient text that I am aware of ever links Polykleitos’s Canon of Proportions to the Golden Ratio in any way. Furthermore, the proportions of the surviving copies of Polykleitos’s sculptures do not correspond to the Golden Ratio. I think we can safely say, then, that Polykleitos’s Canon of Proportions was not based on the Golden Ratio.

One thing Polykleitos’s Canon of Proportions does demonstrate, however, is that numbers and measurements really were very important to at least some ancient Greek sculptors. Greek sculptors and architects may not have been obsessed with the Golden Ratio, but it is quite possible that some of them may very well have been obsessed with mathematics in general.

ABOVE: Roman marble copy from the Naples Archaeological Museum of the Doryphoros, or “Spear-Bearer,” a larger-than-life bronze sculpture of a nude male warrior bearing a spear that was originally cast sometime around 440 BC

Mathematics and the Parthenon

Indeed, not only were numbers and mathematics important in ancient Greek art in general, they were even important in the designing of the Parthenon. The Parthenon may not incorporate the Golden Ratio, but it certainly took a lot of impressive math to design that building.

The architects who designed the Parthenon were these two fellows named Iktinos and Kallikrates. They incorporated all kinds of optical illusions into the building’s design in order to compensate for the effects of foreshortening and perspective. Even though the Parthenon looks to the viewer as though it is full of straight lines and right angles, it actually does not contain a single straight line or right angle at all.

You see, to compensate for foreshortening, the columns of the building are ever so slightly more slender at the tops than at the bottoms and they swell in the middles, making them look straight. Meanwhile, the steps leading up to the building swell in the middle so that they look straight. In actuality, though, if you set your hat at one end of the step, go to the other end, and peer over the edge of the step, you won’t be able to see your hat, because the swell in the middle of the step will be blocking your view.

Conclusion

Ancient Greek sculptors and architects, especially the designers of the Parthenon, were absolutely brilliant. Nonetheless, the notion that the ancient Greeks—or any other people from the ancient world for that matter—were in any way “obsessed with” the Golden Ratio is a misconception. Quite simply, at least as far as we know, the Greeks of the Classical Period did not use the Golden Ratio in their art, partly because there is a good chance they did not even know about its existence and partly because the Golden Ratio is not actually uniquely aesthetically appealing.

There is a small degree of truth here in that ancient Greek sculptors and artists really did rely on mathematics to an extent to enhance their buildings. The Greeks, of course, were not the first ones to use mathematics in architecture in this way; mathematics has been used in architecture since at least the time of the earliest Egyptians and Sumerians. I do not think that really matters all that much, though, in the grand scheme of things. What really matters is that we have these sculptures and monuments to admire in the first place. When admiring the Parthenon, does anyone really think about the ratios that make it up?

ABOVE: Photograph from Wikimedia Commons of the Parthenon at night on the Athenian Akropolis