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Monday, July 11th 2016

0:48

The Parthenon and the Taj Mahal

  • STATE OF EXISTENCE: ready for a break

The Golden Ratio in Ancient Architecture

golden triangleIt’s time to come back to the point of this series. Its topic is the beauty of the number that is usually called phi (ϕ). Given all of my stacking and popping, I think a little recap is in order:

Contrary to the things you frequently may read, phi is not derived from the series of Fibonacci numbers, though they do converge to the value of ϕ: 1.61803…. Its origin lies in the geometry of a pentagon from which we can derive a “golden triangle,” which is distinguished by the fact that the ratio of one side to its base, ...

side over base

... is equal to the ratio of the side and base combined to a side to the base.

side plus base

So that:

golden ratio.

If we want to express this relationship with regard to a straight line, we can say that there is a line, connecting points A and C, running through point B,

golden ratio

and the ratio of BC to AB is the same as the ratio of AC to B, namely the famous phi: 1.61803 ….

I recounted some fascinating properties of phi, and showed you a few interesting features of phi in relationship to the Fibonacci series. After an excursion on some wonders in π, I asserted that, for Christians, the presence of the Fibonacci numbers in the universe and the beauty within the world of numbers itself should lead us, without hesitation, to affirm the wonderful hand of God displayed in his creation. Then I set out on a long excursion on some scientists and mathematicians who see the world of numbers as divine, but, not wanting to acknowledge the God of the Bible, find God in some unexplained and inexplicable manner within his creation itself.

Given the length and seriousness that this series has taken on, I will flesh out the earlier section on the Fibonacci series in nature. For now, I would like to address two questions that seem to go hand in hand.

*****

We find beauty in many work of art. Some works may be more beautiful than others. Many people say that the perception of beauty may in certain cases be due to the fact that artists have incorporated the golden ratio in their creations. Thus we have two considerations to address:

1. Do we actually find the golden ratio in some of the works that are usually cited as examples of manifesting ϕ?

2. Does the presence of the golden ratio actually trigger our response to consider some things as beautiful?

We need to address the first question first because that's inherent in being the first question.

Artists are free to incorporate the golden ratio in their works to their hearts’ content. And if we find beauty in their production, so much the better. A good example is Salvador Dali’s painting, “The Sacrament of the Last Supper.” Its proportions in internet reproductions appear to be pretty close to ϕ, and I can’t be sure how much may have been lost in either trimming or framing. In this case, a tolerance of a couple of millimeters or pixels can be taken allowed.

Salvador Dali Last Supper

For the reproduction I have picked out of the many on the web, this one comes out at 1 : 1.58. Dali leaves no doubt about his intentions, seeing that he inscribed his painting with a dodecahedron, whose twelve sides consist of pentagons.

Livio Golden RatioApparently the two most frequently used illustrations of the golden ratio are the chambered nautilus and the Parthenon, the ancient Greek temple devoted to the virgin Athena, located on the Acropolis. We already mentioned that the nautilus increases his cells in the shape of a logarithmic spiral, though the ratio is not ϕ. Nevertheless, it is frequently used. Even the front cover of Livio’s book, in which he clearly states that the nautilus is not an example of the golden ratio, greets us with a representation of this misunderstood mollusk. You see that authors are frequently less in control of their books than one might imagine.

The Parthenon is often used to illustrates how an architect employed the golden ratio to endow his work with beauty. It appears to be almost a given that you can find the golden rectangle in the Parthenon. But we must ask, where exactly in the Parthenon do we find the golden ratio? People making this claim usually illustrate it, and the lines that are drawn differ from person to person. Precision is important. When it comes to pictures of buildings, we’re not dealing with millimeters, but with much larger entities.

Here's one example from the webpage culturacolectiva.com. It finds not just one, but two golden rectangles in the building, situated next to each other, inscribed by a golden spiral. Note how the smaller one on the left goes down to an arbitrary line on the ground.

culturacolectiva

We can contrast that depiction with what we see on the site Design by Day™ Everything you need to know about the Golden Ratio in Graphic and Design.

Angela Roche Graphic

As most of these pictures do, it extrapolates to the top of the pointed façade, which is lost. The bottom goes to somewhere in the rubble at the bottom of the stairs. The two sides drop in alignment with the eaves and connect to the base in no-where’s land, cutting the platform where the rectangle requires it, but where there is no architectural indication for it.

Similarly, a site on Greek and Roman art, seems to give priority to the rectangle at the expense of architectural detail.

wikidot1

The location of points A and C in this picture are crucial to finding an additional golden triangle, but don’t seem to play a role in the actual construction of the building. Here is another picture on the same site. Note the pronounced leftward shift of the rectangle.

wikidot2

On Pinterest I ran across this little gem:

From Pinterest

The angle at the top is a little more obtuse than in some of the other pictures that extended the lines, and so the rectangle does not need to be as tall to meet the proportions. The base line is located above the stairs. The line is straight, but the placement of the columns is not, except for the front two. And again, the location of the two bottom corners is established by the geometry of the golden rectangle, not by any mark in the building.

We can find the following entry in the Nexus Network JournalTM:

From Nexus Journal

If I read the context correctly, this diagram is intended to make the point that there is no single way of inscribing the golden rectangle in the Parthenon. And that observation obviously negates the idea that phi is incorporated into the Parthenon, and that we see its beauty because of the golden ratio.

Finally, Gary Meisner, on his website ϕ = Phi ≈ 1.61803 comes up with yet another placement of the rectangle, cutting off the eaves.

Gary Meisner

But he expresses some hesitancy about imposing the golden ratio on the temple front as a whole, because it seems to require too many arbitrary decisions. He then seeks for it in parts of the structure, but, even if that should work, it's out of keeping with the conventional belief that the entire facade attracts us with the golden proportion, and so we'll skip that exercise. 

I think that my point should be pretty obvious by now: Lots of people agree that the Parthenon receives its beauty at least partially from the golden rectangle. But there is no unanimous agreement where precisely it is located, a fact that at a minimum should make us a little doubtful concerning this idea. I don't think that beauty is entirely in the eye of the beholder, but the golden rectangle might be. 

Actually, a couple of weeks ago, when I was thinking about this topic as a future entry, I decided to try my hand at the process without knowing what to expect. I knew what would be coming with the Parthenon, but I could not recollect ever seeing any pictures of the Taj Mahal in connection to the golden ratio. There are many of them; I had just not seen them. Well, I have a few good pictures of the Taj Mahal that I took in 2006, so I started to select various rectangles on the building and measured their lines in pixels to see if I could get close to a proportion of phi anywhere. The obvious place to start was with the entryway, trying to be as careful as I could, but also being open to the possibility that I could make an error in placement. Nothing I tried there worked out, and neither did the window-like openings.

Taj Mahal August 2006

 

 

Taj Mahal Taj Mahal
Taj Mahal Taj Mahal

I finally decided that I could not impose the golden rectangle anywhere on the Taj Mahal without fudging.

Then, in the course of looking for good pictures of the Parthenon, I came across my first Taj Mahal/Golden Ration depiction. I did not write down the URL at the time (or I misplaced it), but I remember exactly what it looked like, particularly because it was indicated in exactly the area I had been searching. My measurements confirmed that here was, indeed, a genuine golden rectangle--or so it seemed.

Taj golden

But then a closer look revealed that the person who posted that picture had been creative in placing the rectangle without taking the decorations into account. His or her top line is clearly above the somewhat lighter frame, but then the sides run along its inside. I thinned out the line a little bit compared to what I originally posted last night and highlighted the frame in order to make the obvious fudge more clearer for anyone who might have not seen it. I think I'm just as happy I couldn't find my way back to that particular URL because that person might just be angry with me. Some people ascribe mystical, supernatural powers to phi and the golden ratio, and probably aren't happy when an obvious fudge comes to light. (If I do find it, I will post it, so you don't have to take my word for it. )

Taj goldenTaj golden

The story of phi and the Taj Mahal resembles that of the Parthenon. "Everybody" insists that it's there. "Nobody" can agree where exactly it is, and there is a lot of fudging going on. I'm not saying that there are no instances of finding the golden ratio in art. But I'm skeptical about how strongly it is represented in some of the traditional supposed examples. I have maintained that there is beauty in numbers; I'm not so sure that works of art depend on numerical values for their beauty. More on that next time.

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