Part 2: The Golden Mean and Leonardo

Yesterday I challenged the cherished notion that ancient architects used the golden mean as a design template for the Parthenon of Athens. (For those who don't know, the golden mean is the ratio of 1.618.../1. It also goes by other names: the "golden ratio," "golden section," "phi," or the symbol "ϕ".)

Today let's consider whether Leonardo Da Vinci used this mathematical principle in his artwork. The claim that he did so appears in everything from modern how-to books on composition, to art school lectures, to popular novels such as Dan Brown's The Da Vinci Code.

Leonardo did several drawings of the idealized human figure set inside a geometric grid, including the so-called Vitruvian man.  

The drawing can be overlaid with golden mean measurements, and they seem to click. The distance of the full height of the figure compared to the distance from the ground to the navel is roughly equal to phi.

Leonardo mentioned phi in his notebooks. He illustrated a book called "On the Divine Proportion" by Luca Pacioli. In that book, Pacioli discussed the golden mean and its application to geometrical shapes and the human figure. Leonardo's illustrations for the book mainly include geometric solids such as Rhombicuboctahedron below.


But according to George Markowsky, "the biographies of Leonardo by Clark, Vallentin, and Zammattio et al give no indication that he used the golden ratio in paintings or drawings not intended for Pacioli's book." Instead, both Pacioli and Leonardo himself advocated a Vitruvian system of proportion, using relationships of whole numbers such as 1:2, 1:3, and 2:5.

In Leonardo's own notes to accompany the Vitruvian man drawing, he cites whole number relations such as: "a palm is four fingers; a foot is four palms, a cubit is six palms, four cubits make a man," etc. In the measurement markings on the drawing, he also places whole number ratios (or Vitruvian) divisions, such as halfway to the crotch, etc.

Artists today are familiar with whole number relations in figure drawing, such as "a figure is about eight heads tall," or "the eyes are halfway down the head," knowing that individuals can vary widely from the ideal.

The analysis of the geometry of the Vitruvian man drawing gets really arcane and will probably be endlessly debated. But for the purpose of this post, we have to ask the simple question: If Leonardo was thinking about the golden mean in the Vitruvian man drawing or any other work, why didn't he clearly demonstrate his intention anywhere in his notes?

The golden mean relations that people have found in the drawing ex post facto are not conclusive proof that Leonardo was thinking of phi, because anyone could overlay the figure in other ways with segments exhibiting nearly any other ratio. We would need to find, as Antonio said yesterday in the comments, "historical documents that proved the intention was there."

Was the golden mean a special, divine, or magical principle to Leonardo? Was it a secret aesthetic principle that, like the name of Voldemort, was too powerful to utter? Or was it for Leonardo just one of many fascinating irrational math numbers, such as:

pi=3.1415926535....
√2=1.41421356237....
phi= 1.61803399....

ζ(3)= 1.2020569031....
γ=0.5772156649....

I would like to keep an open mind about all this, especially because we're talking about a fascinating genius who combined art, math, and science in such unexpected ways. But I'm also skeptical of casual claims made about the golden mean geometry in Leonardo's painted work. Even the proponents don't agree in their diagrams, and each diagram on its own doesn't even make sense most of the time.

I bring all this up reluctantly and with respect, because many of my friends and colleagues—many of whom are great painters—use the golden mean centrally in their work and their teaching. My intention isn't to run around upsetting pretzel carts. And as I said yesterday, if any system helps you paint or observe better, than by all means use it. 

And I'm certainly not against the idea of mysticism in art. Much of my own artistic inspiration comes from sources that I can only describe as mystical. What I object to is pseudoscience and misinformation, assertions of fact that have no grounding in science or history.

The story continues tomorrow. 

Wikipedia on Vitruvian man.  

What did Vitruvius and Da Vinci actually have in their notes?

The Myth That Will Not Go Away by Keith Devlin
Misconceptions about the Golden Ratio by George Markowsky
Debunked Legends About the Golden Ratio by Julia and Jesse Galef
Meandering Through Mathematics by Phil Keenan, Ph.D.

GurneyJourney series: Mythbusting the Golden Mean

Part 1: Debunking claims about The Parthenon

Part 2: The golden mean and Leonardo
Part 3: How the golden mean caught on with artists
Part 4: The golden mean and the human body
Part 5: Last question about the golden rectangle