Wednesday, 27 February 2019

The golden ratio

‘Ancient architecture and art are full of examples of the golden ratio’ is a myth that we should blame on Walt Disney. He didn’t invent it, but he sure did popularise it.

[Edited: this originally read that they’re ‘chocker with examples of the golden ratio’ -- but it turns out the kiwiism ‘chocker’ doesn’t cross oceanic boundaries well.]

In 1959 Disney released a half-hour educational cartoon starring Donald Duck, Donald in Mathmagic Land. For decades the cartoon was shown to maths classes in thousands of schools. I saw it at my school in New Zealand in the 1980s. For a good while, I believed its claims -- even though I only half-remembered them.
Donald in Mathmagic Land (Disney, 1959)
Here’s a sample:
To the Greeks, the golden rectangle represented a mathematical law of beauty. We find it in their classical architecture. The Parthenon, perhaps one of the most famous of early Greek buildings, contains many golden rectangles.
-- Donald in Mathmagic Land (Disney, 1959)
The cartoon also states that the golden ratio can be found in pentagrams, and that it can be found in naturally-occurring pentagonal and spiral shapes. The thing about pentagrams is absolutely true, and there’s some truth to the claims about pentagons -- but natural spirals are much more diverse than Donald Duck led us to think. And as for the golden ratio in architecture ...

Mathematical explanation

The ‘golden ratio’, also known as φ, is equal to (√5 + 1)/2, or 1.61803...
A golden rectangle with dimensions 1 × φ. The gold-coloured region has the same proportions as the larger rectangle. If you continue to cut off squares, the remaining rectangles will still all have the same proportions as the original.
The golden ratio is defined as follows. If you have a rectangle with sides 1 × φ, you can chop off a 1 × 1 square and the remaining smaller rectangle will have exactly the same proportions as the original one. This will only work if the original proportion is exactly φ. Such a rectangle is commonly known as a ‘golden rectangle’.

You can use golden rectangles to construct other ‘golden’ shapes: a ‘golden angle’, at the angle of a golden rectangle’s diagonal, and a ‘golden spiral’ like the one shown below superimposed on a nautilus shell.
Nautilus shells famously follow a golden spiral ... except, um, they obviously don’t.
The myth is that golden rectangles pop up all over the history of art and architecture, and golden spirals pop up all over nature. There are elements of truth to this. But they aren’t remotely as common as you might imagine from watching Donald Duck, or from reading Wikipedia’s ‘list of works designed with the golden ratio’.

φ also has some interesting numerical properties:
  • φ – 1 = 1/φ, and φ + 1 = φ2.
  • The first of these equations is simply a restatement of the definition of the golden ratio (see diagram above). From it, we can extract the quadratic equation φ2 – φ – 1 = 0. Solving this gives the value φ = (√5 + 1)/2.
  • In the Fibonacci sequence, each number is the sum of the previous two numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The longer the sequence goes on, the closer the ratio between each number and its predecessor gets to φ: the ratios go 1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, and so on.
  • Powers of φ are closely related to the Fibonacci numbers. If we define Fn = the nth number in the Fibonacci sequence, then
    • φ2 = F1 + φF2
    • φ3 = F2 + φF3
    • φ4 = F3 + φF4, etc.
These mathematical claims, at least, are absolutely true, and there’s a lot more we could add. It’s when we get to the physical world that the problems begin.

The problem

The golden ratio isn’t nearly as omnipresent as its fans would have you believe. You will find φ in some natural phenomena that involve pentagonal shapes, or a repeating growth process. That’s because these things are directly related to the mathematics of φ. The Fibonacci sequence is a recursive growth process, so Fibonacci numbers do pop up in nature, and as we saw above, the Fibonacci sequence generates the golden ratio.

But it definitely doesn’t happen everywhere. In particular, nature does not favour golden spirals. There are other logarithmic spirals in nature -- nautilus shells are the best known example -- but only a spiral at a specific angle is a golden spiral. Even in situations where Fibonacci numbers arise, like clustered leaf arrangements on a plant stem, the spirals aren’t golden spirals.
NGC 232: no golden spirals in sight. If you get the spiral arms to match the curve at the top and right, then they are obviously inaccurate at the left and bottom, and in the centre.
And then there’s art and architecture. Here, you have to look really hard to find the golden ratio. In ancient Greek art and architecture you won’t find it at all. Unless you fudge it.
Fudged golden rectangles. From top left: caryatids on the Erectheium, Athens; Leonardo’s ‘Mona Lisa’; a live human woman (all from Donald in Mathmagic Land, 1959); the Parthenon (from this webpage). What are the drawn rectangles even supposed to demonstrate? That you can draw rectangles on pictures? None of the Disney ones match anything in the images. In the Parthenon picture the top edge matches the building, but the left and right edges are only approximate, and others just show the theory’s falsehood: the bottom edge of the largest rectangle, and the right edge of the largest square, don’t match anything on the building.
Sure, you’ll find websites all over the place claiming to find golden rectangles in all sorts of places, especially the Parthenon. They’re heavily flavoured with conspiracy-theory-style thinking. Some people can get very, very angry if you express doubts. The talk page on the Wikipedia ‘list of works designed with the golden ratio’ is interesting reading. In 2008 there was a minor war over the subject: there’s one person patiently and doggedly requesting substantiation, details, and documentation, while others -- one person in particular -- get increasingly frustrated. The reason they’re frustrated is because they can’t find any decent substantiation. And the reason they can’t find it is because it doesn’t exist.

It can sometimes be a good joke to satirise some of the claims. Here’s a page from the webcomic xkcd that superimposes golden spirals over anything and everything. You can draw rectangles and spirals anywhere you want ... it doesn’t mean that they’ll fit anything.

Let’s move on to some specifics.

Myth 1: The Parthenon is designed around φ

This is probably the most popular golden ratio myth. The Parthenon is the famous temple of Athena in Athens. Across the internet -- and in Donald in Mathmagic Land -- you’ll see many images of the Parthenon with golden rectangles superimposed on various bits of its facade.
Donald in Mathmagic Land (1959) uses a hand-drawn Parthenon. Not too surprising, then, that the fit is so tidy.
If you do this with an accurate elevation plan, though, you’ll quickly find that golden rectangles don’t actually fit any edges on the building. 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.

Here’s one diagram that depicts the Parthenon with measurements full of various multiples of φ, π, and e. A few problems:
  1. The measurements are all wrong. For accurate figures, see Orlandos (1976-1978). Selected measurements are also quoted by Lehman and Weinman (2018: 167-168).
  2. If you’re giving examples of the golden ratio and you have to resort to proportions like φ3√5 and 10π/3, you’re doing it wrong.
  3. The ancient Greeks didn’t know the values of φ and π to any great precision. There’s no evidence anyone even knew of φ until Euclid. As for π, Archimedes calculated its value precise to two decimal places two centuries after the Parthenon was built; in the earlier period, the best approximation of π would have been that of Antiphon, who calculated only a lower bound for its value, and was doubtless less accurate. And the ancient Greeks had no clue what e is, because they hadn’t invented logarithms or compound interest: e wasn’t defined until the 1600s.
[Addendum, a couple of days later: I spoke rashly in point 3. φ probably was known to mathematicians of the late 5th century BCE. Important points about the icosahedron and dodecahedron appear in book 13 of the Elements, which owes a lot to, and may even be largely copied from, Theaetetus of Athens, a key early figure in the study of irrational numbers. The point about precision stands, though.]

Here’s another site that looks at a whole bunch of supposed golden rectangles in the Parthenon facade. Its conclusions are negative, but in my opinion not nearly negative enough.
A photo used on Claim 1(a), below, relates to the yellow rectangle, and claim 1(b) to the red rectangle.
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%.
Actual proportions of Parthenon metope + triglyph (west facade dimensions), with superimposed golden rectangles in red (the correct height) and blue (the correct width).
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).
Photo of the Parthenon from this webpage: the green rectangles are original, the red rectangle added by me. The green rectangles supposedly show golden ratios all over the place. The red rectangle is a real golden rectangle. It doesn’t fit.
You might reply that these are near enough: that the intent was to produce golden rectangles, and the inaccuracies are just the result of imperfect building techniques.

You could argue that. But only if you ignore the fact that the Parthenon is actually rather well engineered. The precision is way better than one part in a hundred. Remember how the metope:triglyph ratio is within 0.25% of the intended proportion, 3:2 (myth 1(a), above).
The Parthenon, reconstructed, with superimposed golden rectangles and golden angles all over the place. None of them come even close to fitting anything. This elevation was drawn up by the architects James Stuart and Nicholas Revett in the 1750s: I use it here, rather than a photograph, to avoid foreshortening. (Source: Stuart 1787, chap. 1 plate 3)

Myth 2: The sculptor Pheidias used φ

This myth is closely allied to myth 1, because Pheidias was credited for the colossal statue of Athena Parthenos in the Parthenon. Taken in conjunction, they’ve often ended up making Pheidias the architect of the building (he wasn’t) as well as a sculptor.

[Addendum, a couple of days later: I should have qualified this. Pheidias was the supervisor of the Parthenon project. But the architect was a different man, Ictinus, who also designed the extraordinary temple of Apollo at Bassae, and had a hand in the Telesterion in Eleusis and the Periclean Odeon in Athens. He was a very skilled architect.]

The myth about the sculptures was made up in the 1910s. It happened hand-in-hand with choosing the letter φ to represent the ratio. According to Theodore Cook, the letter φ was suggested by the engineer Mark Barr
partly because it has a familiar sound to those who wrestle constantly with π (the ratio of the circumference of a circle to its diameter), and partly because it is the first letter of the name of Pheidias, in whose sculpture this proportion is seen to prevail when the distances between salient points are measured. So much is this the case that the φ proportion may be fitly called the ‘Ratio of Pheidias.’
-- Cook 1914: 420
The idea of φ as a counterpart to π is reasonable. The stuff about Pheidias, though, is pure fiction. We don’t know the proportions of Pheidias’ free-standing sculptures, for the simple reason that none of them survive. We have many of the decorative sculptures on the Parthenon, but they don’t exhibit the golden ratio so as you’d notice. We do have descriptions of some of Pheidias’ statues, but the descriptions don’t discuss any ratios, let alone the golden ratio.

It’s not clear whether the myth was invented by Barr or by William Schooling, the person that passed Barr’s suggestion to Cook. Apparently in 1929 Barr stated that he didn’t ‘believe’ Pheidias actually used the golden ratio, but I haven’t managed to get hold of the later article to read what he actually says there.

Myth 3: Plato’s divided line, something something

Plato’s analogy of the divided line (Republic vi.509d-511e) chops up the world into the physical and non-physical realms, which are then each divided up into two sub-sections in the same proportion.
It has nothing at all to do with the golden ratio. I bring it up here because Plato talks about the visible and intelligible realms being subdivided in the same ratio as the overall division, and apparently some of Plato’s readers are unable to imagine this happening with any ratio other than φ.

Myth 4: Vergil’s Aeneid uses φ

This one actually originates with a classicist, George E. Duckworth. He argued it in a series of articles and a 1962 book. Hardly anyone took it seriously at the time -- see the reviews by Dalzell and Clarke -- and no classicist takes it seriously nowadays.

Duckworth assumes that Vergil knew the numerical value of φ, knew the Fibonacci sequence, and understood the relationship between them. He then identifies examples of the golden ratio in passages with relative lengths anywhere between 1.5 and 1.75, and in passages whose length in lines is a Fibonacci number.

Fibonacci numbers, unfortunately, weren’t known in Europe until Fibonacci wrote about them in the 1200s. Their connection to φ wasn’t known, or at least not widely known, until Simon Jacob noticed it in the 1500s. So imagining Vergil using these ideas is ... difficult.

The reviews linked above also point out copious examples of how Duckworth cherry-picks his data, massages it, and conceals imprecisions. Clarke takes the additional step of illustrating that arbitrary ratios can be found in any poet if you look hard enough, by analysing a poem by John Betjeman in the same way. (He picks Betjeman, ‘a poet certainly oblivious of the Golden Section’, because his style is antithetical to the abstract; perhaps also because of Betjeman’s documented incompetence at maths and laziness as a student.)

Myth 5: ‘European paper sizes’ -- A4, A3, etc. -- are golden rectangles

Yes, I really have seen people claim this. This one is a twofer:
  1. Those paper sizes aren’t European, they’re the ISO international standard.
  2. The actual ratio of A4/A3/etc. paper size is √2 (1.414...), not φ.
(Actually the paper size closest to a golden rectangle is US legal: 215.9 × 355.6 mm, a ratio of 1.647. And legal looks weird. So much for golden rectangles being the ideal proportions.)

Myth 6: If you ask people to pick a random number between 1 and 100, they’ll prefer 61 and 37 because of φ

The idea here is that the human brain is naturally attracted to the golden ratio. There aren’t any well tested scientific studies showing that, though. I’ve seen someone seriously claim that these choices are hardwired into the human brain because 61 = 100/φ and 37 = 100/φ2. (Why on earth would our brains care about 1/φ2?)

It certainly seems to be true that people choose odd numbers, prime numbers, and numbers ending in 7 or 3 extraordinarily frequently when asked to pick a number randomly. I haven’t managed to find any scientific studies on this either. Some informal surveys that I’ve found (1, 2, 3) don’t bear out the 61 claim at all, and the 37 claim only inconsistently.

But it does seem to be the case that when people choose a number from 1 to 10, by far the most frequent choice is 7; when they choose a number from 1 to 20, they’ll pick 17 as much as 20% of the time. For some reason, though, golden ratio fans don’t mention these two phenomena so much. I guess it’s too obvious that they have nothing at all to do with the golden ratio.

In any case the calculations are wrong. 100/φ is 61.803..., that is, closer to 62, and 100/φ2 is 38.197..., not 37.

Myth 7: Leonardo da Vinci’s drawing ‘Vitruvian man’ uses φ

It doesn’t. This article by Takashi Ida does a detailed investigation of some claims and possible uses of φ in the drawing, and none of them are true. In particular, Ida shows that the ratio of Leonardo’s circle to his square is about 0.606 to 0.609, rather than 1/φ = 0.6180..., a difference of 1.51%; and he argues from the marks Leonardo placed on the diagram that the ratio he intended to use was precisely 137/225, or 0.6089, which corresponds well to the measured ratio of the circle.

In closing ...

I’d better stop: I’ve gone a long way off-topic from Greek architecture anyway.

There are some genuinely interesting things about the ‘golden ratio’. It does have some pretty interesting numerical properties. Several proofs in Euclid’s Elements book 6 do deal with φ in one way or another. He calls it ‘the extreme and mean’: the phrase ‘golden ratio’ wasn’t invented until the 1800s.

And it’s true that φ can be found embedded in the diagonals and other ratios of several geometrical shapes. In a regular pentagram, each vertex and intersection has two adjacent line segments with lengths in the ratio φ. Or, put another way, the diagonals of a regular pentagon intersect to create line segments with the ratio φ. As a result, any geometrical structure with pentagonal features is going to feature φ in some way -- including two of the ‘Platonic solids’, the dodecahedron and icosahedron, as well as the areas of the tiles in Penrose tiling.

And some artists and architects have definitely used φ in their work. The Swiss-French architect Le Corbusier based a design system on φ and the Fibonacci numbers in the 1940s. (Whether this has anything at all to do with the supposed golden rectangles on the UN headquarters building in New York is another matter.) Salvador Dali’s Last supper definitely takes inspiration from the mathematics of φ: the canvas is within 1% of being a perfect golden rectangle; the figures are in groups of 2, 5, and 13 (all Fibonacci numbers); and the painting is dominated by a dodecahedron (remember pentagons feature φ heavily) whose design is modelled on one of Leonardo da Vinci’s illustrations for Pacioli’s Divina proportione (1509), the book that kickstarted the modern interest in φ.
Salvador Dali, The sacrament of the last supper (1955)
But other than really blatant cases like these, I recommend treating claims of the golden ratio in art and architecture with great suspicion. Golden ratio fans are wont to interpret any old proportion as a golden rectangle, to gloss over imprecisions, and to make completely fictional claims about the history of the ratio. Don’t ignore the fact that there are other ratios in the neighbourhood of 1.6. Always be alert for cherry-picking.



  1. Who knew?! Pi has a golden ratio and Pi defines Quadrature!

    Golden Rectangle of Quadrature (diagonal = circle’s diameter, ratio of long side to short side):

    Qphi = Phi of Pi = sqrt(Pi)/sqrt(4-Pi) = sqrt(Pi/4-Pi)
    = 1.9130583802711007947403078280203..

    Pythagorean Quadrature (hypotenuse = diameter,
    ratio of hypotenuse to long side = Qcue):

    Qcue = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2) = 2(sqrt(1/Pi))
    = 1.1283791670955125738961589031215..

    "The circle cannot be squared,
    but squared circles exist ... apparently."

    1. Quick summary:

      Given: Circle-squaring right triangle
      inscribed in a circle where D = 2
      and a = sqrt(4-Pi), b = sqrt(Pi), c = 2

      Since a^2 + b^2 = c^2, 4-Pi + Pi = 4,
      and 2 = 2. Who knew?! Pythagoras?

  2. I did my own article about the ancient Greeks and the Golden Ratio in November 2019 in which I summarized a lot of the main points you made in this article, but also looked a bit deeper into the history of this misconception.

    The modern obsession with the Golden Ratio seems to mostly originate from the eccentric German psychologist Adolf Zeising (lived 1810 – 1876), who was thoroughly convinced that the Golden Ratio could be found virtually everywhere in nature and in art. In a book published in 1854, Zeising called the Golden Ratio "das Grundprinzip aller nach Schönheit und Totalität drängenden Gestaltung im Reich der Natur, wie im Gebiet der Kunst enthalten ist."

    Naturally, Zeising's writings set off a whole flurry of other publications speculating about the Golden Ratio and its supposedly ubiquitous presence in art and nature. This basically seems to be where the idea of the ancient Greeks using the Golden Ratio for everything mostly originated from. Pacioli was an important precursor to Zeising, but Zeising is the one who seems to have really set things off.