Wednesday, January 16, 2013

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:

phi= 1.61803399....
ζ(3)= 1.2020569031....

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.  


  1. Leonardo was a competent technician. But his artistic legacy is vastly overrated, and he is a super-magnet for crackpot theorists.

  2. The kicker for me in debunking the application of the Golden Mean to the Vitruvian man is that, as you mention, Leonardo was such a fanatic note taker, and (to my knowledge) his notes on the Vitruvian man include no mention of phi (by any of its names).

  3. Excellent post. Next up in the myth busting department: The Hockney/Falco optical theory?

  4. Very interesting, thank's. But it seem's that there is a little mistake in the formula on the Joconde's picture. The correct formule should be : 1/x = x/(x+1) that makes x2 - x = 1
    don't you think so ? ;-)

  5. Now you've gone and done it...upsetting pretzel carts everywhere!

  6. And if we put in the mixture some cropping of the original Giocconda, then the Golden Mean may seem more unatainable (or not, who knows)...
    I always heard the original picture had some columns, but I couldn't find my original source.
    So, some Wikipedia here:

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  9. Great series of posts this is becoming. I myself was not comfortable with these theories .. too much analysis.

  10. You do a good job of exposing the persistence of logical fallacies in painting theory. Especially theories that cherry pick facts, and rely on proof through verbosity. But, as you point out, if the use of a logical fallacy still results in a good painting, then what is the harm? But if it results in intimidating a student of painting to feel inadequate, or somehow too ignorant to make a "proper" painting then I guess that is a harm.

  11. Leaving aside the debates on whether or not the golden ratio is present in the work of certain artists or particular pieces of art, I am especially interested in why we supposedly perceive this ratio as attractive. Is there a scientific explanation as to how we think of the proportion as beautiful?

  12. JC Barquet, that is a very important question, and I'll get to it the day after tomorrow.

    1. Hi James, when are you going to write a book on composition to compliment your excellent colour and light?

  13. The persistence of certain ratios throughout nature would suggest a sort of singular source behind it all. How better to make great art than to mimic the methods of the great creator?
    To debunk the assumption of the natural occurrence of the ratios in nature as well as their artistic application would be refreshing.

  14. In art school I understood that the Golden Section ratio was settled upon because it was the 'natural' way most people tended to divide up space. Maybe that is bunk, too. Never consciously use it.

  15. Fascinating topic, JG. I'm naturally a skeptic as well, but was shown an amazing demonstration by a painting professor at B.U. (Richard Raiselis) of the Golden Mean used in Hopper's "Night Owls". He projected a slide of the painting on the wall and proceeded with a yardstick and string to show us how various angles and verticals lined up perfectly with each other using phi (and some other things I can't remember). Shall I introduce the two of you online?

  16. And then there’s the “poor old” Nautilus…

  17. Excellent post! Great to see someone rationalize about such an established idea. To me this fact alone is great and more important than if it's actually true or not. Everyone can just make their own conclusions.

  18. In my own experience as an architect, I can tell you that you can find the golden mean if you look for it hard enough. Much like interpreting the predictions of Nostradamus, if you try hard enough to make something fit, focus on the elements that support your case, and ignore the elements that don't, you can argue for the existence of any number of 'special' relationships in something as complex as a piece of architecture.

  19. Forget the golden mean, and the nautilus, and think for a moment of the pretzel which has a neat shape too. And if you pick up a pretzel after it's knocked out of the cart, and wipe off the dirt, it tastes pretty good.

  20. I think you are actually upsetting pi carts.

  21. Syldem, I think the correct algebraic formula for the golden ratio is "one plus the square root of five divided by two."

  22. Yes Mr Gurney, that's correct but "your" formula is in fact the only positive solution of the equation "x2-x-1=0" and the mistake on the Joconde picture is that we have somewhere (1-x) that gives a negative value. Then we obtain something positive equal to something negative. oups ! ;-)

  23. I just found in fact that the formula in the Joconde gives 0.6180339... that is 1/1.6180339
    so, it's not a mistake it's just the expression of the ratio and not the "Golden number" itself
    Sorry !

  24. Excellent series! I especially enjoyed the source material in the links! The evidence points to confirmation bias - ask the question in the right way and you get the answer you're looking for.

  25. Hi James,

    Kind of a latecommer to your wonderful site. Just a comment about this subject...
    I know the fanatical devotion geometry ("sacred" or otherwise) inspires in many artists and influential teachers.

    The only thing I want to say about this is that I don't think this subject belongs in either the categories of unassailable dogma or "pseudoscience".

    I have looked at many of my photos and drawings from my youth when I knew almost nothing about formal composition and definitely nothing about the golden mean. It "clicks" as you say, more often than not.
    I get that it's one thing to say that the ancients consciously used it, and quite another for it to atually be present in the work.
    If it can be found in nature, and mapped onto the human body under average conditions, individual (uniqueness notwithstanding) it is needn't be deliberate, mystical or (proto/psuedo) science driving it's more than cooincidental presence in art, design and architecture through the ages. It could very well be just be instinctual or subconscious or even consciously or semi-consciously eyeballed - which would account for the slight "fuzziness" in peoples mappings of one onto the other.
    The bias you speak of doesn't have to come from wishful theorists but could be there on the part of the artists themselves.
    Barnstone has a lot to say about root rectangles and he can map that stuff onto art until the cows come home. Did the artists necessarily consciously use them or is it a case that "copying from nature" inserts it into the work and the habits and instincts covertly?
    Who knows?
    It's just me, but I personally wouldn't be so quick to apply a scientific/skeptical bias or label just because there are exceptions and an apparent, as yet, lack of found documentation. This lack can be for the reasons I just proposed OR, if said geometry WAS indeed part of a "mystical" transmission among "initiates" of "mystery schools" they wouldn't have written it down explicitly. To me it doesn't really matter which.
    But, while you aren't necessarily "overturning apple carts" by saying, "if it helps you make art have at it, but it's probably pseudoscience for wishful thinkers" you are still, in my opinion, creating artificial categories of validity.


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