**
Win Corduan**

**PAGE 6: PHI IN ART**

**The Mona Lisa**

If you do an internet image search, you may be totally blown away by where and how the "golden magic" is applied. When you reflect on how many people link it to numerology and hidden powers of perception, it gets just a little bit 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 (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 instances 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. We know that Leonardo knew about the golden ratio, so his use of it may very well have been intentional. It is even possible that he may have thought that using the golden rectangle as a template for Mona Lisa's face would enhance her beauty.

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

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.

**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.

**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.

**The 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 the article "Beauty" in the on-line *Stanford Encyclopedia of Philosophy. *

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

**Michelangelo and the Sistine Chapel**

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 an imaginary 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 if one were to draw a straight line between the edges of the ceiling segment passing through the point where God's and Adam's fingers almost touch, that line is in a golden ratio with the point between the fingers serving as the division between the longer and shorter segments.

I have no quibble with this discovery. In fact, I think it's pretty cool. Apparently Michelangelo knew of the golden ratio and implemented it here in this important part of the ceiling decoration. 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.As one can see in the picture below, there are many other compartments of the same length that are not structured according to the golden ratio. Would I really want to say that they are not as beautiful?

**Phi and the Recognition of Beauty**

It's time that we addressed the general belief that the golden ratio in some inexplicable way catalyzes our appreciation of beauty. Arguments and claims pro and con fly back and forth on this issue. Certain experiments have supposedly shown that this idea cannot be documented with evidence while others, usually less formal ones, seem to support it. I suspect that this matter is similar to a court case where each side brings in their expert witnesses to under-gird their case. Let's keep in mind that we're talking about a very narrow window. **1 : 1 1/2 **is too little; **1 : 1 2/3 **is too much. So personally, I would be surprised if eventually there were some clear and objective proof __for__ the aesthetic appeal of phi in all of it's precision. There's no question that it is a pleasing shape, but when you get down to 1.61803... vs. 1.667 ... vs, 1.6 ... vs. 1.5, I'm not sure that our cognitive apparatus is structured with sufficient precision to hone in on the golden number. Still, it's not something that I want to be dogmatic about, mostly because I can't. The one thing I do want to caution us about is that, as I have insisted all along, is that if phi were to have unusual appeal, that would be the case because the Creator instilled in in our cognitive receptors, no in any numerological or magical power inherent in the number.

Let's play a little game. I really don't even want to call it an experiment because it lacks most everything that validates a test. [Nevertheless, it may be fun how it turns out. The official "test" is over by now, but feel free to take it just to test your own perception.].The table below contains various formats of a picture of a "Cherokee maiden" that I took ...

*[Queue up Asleep at the Wheel]*

... a couple of years ago in Cherokee, North Carolina. Each one of these pictures has a different ratio of its sides; one of them is in the golden ratio. Is there one that strikes you as more beautiful than others? I sure hope so because some of them are pretty badly distorted. Which one do you like best? I shall disclose which one is in the golden ratio at some future time. In the meantime, have fun with it.

1 |
2 |
3 |

4 |
5 |
6 |

I look forward to hearing from you which one you picked.

**Results of the Golden Ratio Game**

Thanks to everyone who gave me input on the Golden Ratio test. Here is a table of the results as of 11 pm tonight. Voting is now closed.

**What does this mean? ** Nothing earth-shaking, given the absence of proper scientific methodology.

- The responders were self-selected;
- The sample was small.
**n=44**, as my statistical friends would say. - I had no control over what you actually saw on your screen, i.e. distortions.
- I have no statistical formulas to evaluate whether the data would be meaningful even if I had used proper methods.
- Theoretically, I should have given you two choices: first and second. As it turned out, it wasn't necessary since it was a neck-on-neck race between #s 3 and 4. But imagine a scenario in which the votes for first place were scattered all over the table, but #2 was pretty much the unanimous choice for second place. We could then flirt with the idea that #2 must be the truly specia onel.
- There's a problem with precision since you cannot get fractions for pixels, at least to the best of my knowledge.
- Another point added by my friend Jimm W.: Even the placement of the pictures on the table could influence which one you choose.

Nevertheless, we can take this "experiment" for what it is, an informal inquiry among friends resulting in a rough overview of preferences. we do see that, among those who took the trouble to respond, there wasn't any mysterious attraction to the in the golden ratio**. **Given these reflections, and, if this quiz has any validity at all, we see that we like the proportion in the range of 1.6, but beyond that , it would be really hasty to draw any further conclusion.

"Can you handle the truth?" as somebody said to somebody else in some movie, whose title I forgot. Of the pictures in the chart, the one that is closest to the golden ratio is number 4. The distinctions have to be quite precise since that's what phi is all about. I must confess that if I had been asked which format I considered most attractive, I probably would have selected no. 3.