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Tuesday, July 26th 2016


Pacioli, Leonardo, and Michelangelo


It's cooled off a bit compared to last week. So, I got a little bit of gardening done yesterday. Also, the swimming pool of Smalltown, USA, is open once again. It was closed all last week because it is located right along the fair grounds, and last week was the 4-H county fair. Neither June nor I were particularly interested in the fair this year, partially because of the heat.

I paid a visit to Dr. M, my skin doctor, today. I expected him to scold me for not wearing long sleeves all of the time or not using sun block enough. He would have been only partially right, had he done so. As it was, he skipped that part and gave me a stronger cream to put on my arms. -- No, I'm going to post any pictures dealing with my skin.

Well, I didn’t get much of a response to my golden-spiraled swan, not that I expected to cause a buzz with it. I had two purposes in posting it. One was to see how one can make things look as though they incorporate ϕ, even for a moment, until one takes a closer look. The poor swan in the picture has absolutely nothing to do with the golden ratio, at least to the best of my knowledge and analysis. With a little adjustment of the picture, I got it to the point where I could make the spiral center focus on his head, and then the body fit pretty nicely into the back, actually. The obvious problem is that much of the left side of that picture contains nothing but water. So, the total dimensions for the picture notwithstanding, the swan has his own proportions, which don't come out anywhere near the golden one, once you let the water drain out. The other purpose I will tell you about some other day.


I am working through and improving the entries in this series on its separate site. In the process I am not correcting the previous blog posts, unless I run across something so utterly egregious that I don't ever want posterity to see it. Please use the omnibus site to catch up or refer back to.

I have really honestly wanted to get to the Fibonacci numbers and ϕ in nature, but as I keep sorting through material on the golden ratio on the internet, I keep running across its applications in art and architecture that really need some commentary. Where and how the "golden magic" is applied is almost scary. It seems as though there is nothing that we might consider to be beautiful that has not been given the ϕ treatment by golden-ratio-enthusiasts (hereafter: GREs).

Golden ratio obsession (as opposed to its simple recognition and use) can be traced to to an Italian mathematician, Fra Luca Pacioli (1447-1517). (See Livio, 128-37, for more on this and the next few paragraphs.) Up until then, the ratio had been called by its original name, "the proportion between mean and extreme." Pacioli wrote a 3-volume work on it, entitled, De divina poportione -- On the Divine Proportion. He saw multiple spiritual meanings in it as something that was both uniquely created by God and expressive of God's nature. However, he did not go so far as to see the golden ratio in every nook and cranny.

Pacioli and Leonardo da Vinci were brought together by fate and politics, and Pacioli taught Leonardo about the "divine proportion" insofar as the latter may not have had not had previous knowledge about it. In return, as Livio (133) put it, Pacioli had the "dream illustrator" for De divina poportione since Leonardo provided the pictures of geometric solids and other graphics for the book. Consequently, there is no problem with looking for the golden ratio in some of daVinci's works and being confident about it if one has found it clearly in certain places. Of course, GREs have spared no effort in finding it in all of Leonardo's pictures, multiple times over in certain cases. Sadly, their unsparing vigor winds up concealing genuine substantiations of ϕ underneath the multitude of invented versions.

For example, efforts to include the Mona Lisa in that group suffer from many of the same problems as those we’ve mentioned before with the Parthenon and the Taj Mahal. People appear to begin by assuming that the golden ratio must be there, but then have to find some place where it might actually fit. For this part, I’m going to inscribe my own drawings on the Mona Lisa, based on what I’ve found on the web. That way, I won't pour any more rain on anyone’s parade in particular. Many versions are just copied and pasted from site to site. If you want to see these and others, some of which are just plain bizarre, look for them on Google or your personal search engine of choice.

After looking at numerous contrived examples, it appears to me that Mona Lisa’s face can actually be very nicely circumscribed by a golden rectangle. It does not strike me as an ad hoc imposition.

Mona Lisa's Face

Fig. 1 Mona Lisa's Face surrounded by a golden rectangle

This picture limits the rectangle to the open face, taking the measurements at the longest and widest extensions. In other words, there is good reason to believe that the dimensions of the rectangle mean something.

One problem is that GREs are not content to find a single likely instance and must build nested rectangles, spiral, triangles, pentagons, pentagrams, and geometric objects Euclid would never have dreamed of that are somehow supposed to contribute to the painting's golden-ness. Obviously (at least to you and me) things don’t work as well with such imaginary placements. Consider the picture below. Again, I have redrawn it based on multiple instances of its appearance on various websites.

Mona Lisa's Face

Fig. 2 Half of Mona Lisa's head decorated with a golden rectangle

What’s up with that? as the saying goes. This rectangle is a little bit larger as a whole than the previous one; it includes the top of the hair. The proportions are correct, but this rectangle doesn’t frame anything on either side. On the left, it loses itself in the countryside; on the right it cuts through her hair, eye and cheek. The reason it is placed there is because it is actually a part of an assemblage of golden rectangles. Thus, we can create a larger golden rectangle by adding a square to the right of the one that's there, though the resulting rectangle also has no clear moorings on the canvas.

Mona Lisa's Face

Fig. 3 An extended golden rectangle stuck to Mon Lisa's head

We can go on from there, if we wish, and make more rectangles. My point is once again that, regardless of whether one can decorate the picture with one or more lines of golden proportion, if they don’t have a direct connection to the work of art, there doesn’t seem to be much point to it, and it becomes doubtful that the artist intended to create that particular pattern. Thus, Fig. 1, seems to reveal a golden rectangle. Figs. 2 and 3 strike me as highly implausible.

Vitruvian Man

Speaking of Leonardo and Pacioli, the figure that led to Leonardo's later drawing of the "Vitruvian Man" is also described in De divina proportione, and it would be easy to conclude that, therefore, he, too, must obey golden proportions. That just goes to show how easy it is to conclude something wrong, and for that idea to take on a life of its own under the guidance of the GRE's. There is an excellent treatment of this drawing by Takashi Ida of the Nagoya Institute.

Why is this famous drawing called the “Vitruvian” man? It is based on a description of human anatomy by the first-century Roman architect named Vitruvius. His description of the human person, endorsed by Leonardo, emphasizes rational proportions and symmetry. It is in a section of the book by Pacioli that explores various kinds of proportions and ratios, not just the "divine" one. 

Did I just say “symmetry”? What an interesting thought! Could it be that symmetry also arouses a sense of beauty in us? Vitruvius thought so, echoing an idea propagated by Aristotle and held to some extent by most writers until golden ratio fever set in.

Symmetry also is the appropriate harmony arising out of the details of the work itself: the correspondence of each given detail to the form of the design as a whole. As in the human body, from cubit, foot, palm, inch and other small parts come the symmetric quality of eurhythmy. [Vitruvius, On Architecture, Frank Granger, trans. (Cambridge: Harvard University Press, 1970), 26-27; cited in the article "Beauty" in the on-line Stanford Encyclopedia of Philosophy. 

St. Thomas Aquinas left it open as to what the right proportion may be in a given instance, 

There are three requirements for beauty. Firstly, integrity or perfection—for if something is impaired it is ugly. Then there is due proportion or consonance. And also clarity: whence things that are brightly colored are called beautiful. [Summa Theologica, vol.I, q. 39, a. 8, cited in Stanford Enc. 

We cannot occupy ourselves with such alternatives for the moment because I must press on.

As far as I can make out, whereas Leonardo golden ratio enthusiasm is a long-standing phenomenon, it is only in its early stages for Michelangelo. Apparently it received its big impetus due to a line discovered on the picture of the creation of Adam in the Sistine Chapel ceiling. (Thanks to David O. for calling my attention to this last year.) It has been discovered that a straight line between the edges of the ceiling segment passing through the point where God's and Adam's fingers almost touch turns out to be in the golden ratio, with the dividing point located right between the fingers.  

Creation of Adam

I have no quibble with this discovery. Of course, the prattle along the line that "now we know why we have always liked that picture because the golden ratio was there even though we didn't know it," is silly. I can't imagine that the picture would be any less appealing if it were shorter by a few inches on one side. For that matter, if I'm not mistaken, there are plenty of compartments of the same length where you can't plausibly stick a golden ratio. So, even though they do not carry the same significance as the one with God and Adam, would I really want to say that they are of lesser beauty?

Next time I really hope for sure: ϕ in nature.

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