Win Corduan
PAGE 5: PHI IN ARCHITECTURE
General Questions
We find beauty in many works 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 allowed. 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. Does the obviously intended conformity to the golden ratio enhance the beauty of the painting? Speaking for myself, I wouldn't necessarily even call it beautiful or attractive. Don't get me wrong; it is fascinating to me in terms of its structure and various symbolic allusions, and I'm not repelled by it. For all that I know, given a choice of various formats, I might pick this one as the one that that appeals to me the most, or I might not. I simply do not know. I definitely don't feel any mysterious attraction to this painting, just as I would not be drawn to a cubist work by Picasso, in contrast to a picture by Marc Chagall, whose paintings enchant me. (I defy anyone to try to find the golden ratio in the picture below.)
The Parthenon The Parthenon is often used to illustrate 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, but the lines they draw vary, often drastically, 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, so, even though we have to give ourselves leeway when it comes down to counting pixels in a reproduction, the actual lengths can't be off by meters and still count. Also, take a look at the placement of the spirals in these pictures. According to proper golden ratio theory, the spiral (whose mathematical properties I will discuss later), is supposed to provide a focal point. Golden Ratio Enthusiasts have been know to call it "the eye of God." In any number of illustrations, the spiral focuses on nothing.
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. This whole depiction doesn't make any sense. Originally I had written "to me," but there's nothing personal about it. It can't be that the two rectangles are supposed to add up to a third, larger golden rectangle. Remember that one creates a larger golden rectangle from a smaller one by adding a square to it, not a second rectangle. This combination of two perpendicular rectangles that don't even fit onto the building is nonsense. The spirals are meaningless. 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. 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. Again, the spiral does not lead us to see anything important. Similarly, a site on Greek and Roman art, seems to give priority to the rectangle at the expense of architectural detail. The location of points A and C in this picture are crucial to finding an additional golden rectangle, 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. On Pinterest I ran across this little gem: 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: 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. Gary B. Meisner, who calls himself the “phi-guy, ” maintains a strong web presence devoted to phi and the golden ratio. He is certainly a Golden Ratio Enthusiast (hereafter: GRE), but a rational and discerning one. On his website ϕ = Phi ≈ 1.61803 he comes up with yet another placement of the rectangle, cutting off the eaves. 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 consensus where precisely it is located, a fact that at a minimum should make us a doubtful concerning this idea. If all of us are impressed by the beauty of the Parthenon, and that beauty is supposed to be based on its incorporation of the golden ratio, why do we see the golden ratio in different places (if we see it at all). I don't think that beauty is entirely in the eye of the beholder, but the golden rectangle in the Parthenon very well might be. The Taj Mahal My Futile Quest Actually, a couple of weeks ago, when I was thinking about this topic as a future entry, I was quite familiar with the treatment that the Parthenon has received over the decades by GREs, but I didn't remember anyone claiming to have found the golden ratio in the Taj Mahal. 
The best place to start would with the facade, of course. The two pictures below show what I found:
Taj Mahal Facade 1 Taj Mahal Facade 2
The rectangle in the second picture actually works out within acceptable limits of precision, but only on a two dimensional picture. It has no architectural basis. The rectangle is not actually there because the two smaller domes to either side of the bigger one are recessed, and one's eye would not see a rectangle of golden proportions there, but two two domes above and slightly behind the facade. And, come to think of it now, of all the various attempts of finding the golden ratio in the Taj, I don't remember anyone coming up with this version, most likely because it is clearly bogus and even compulsive GREs draw the line somewhere.
Then I turned to various permutations of the entry way with no luck. Finally I measured the smaller arch-like structures and again found nothing that was architecturally valid and came close to phi in its height/width ratio
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I finally decided that I could not impose the golden rectangle anywhere on the Taj Mahal without fudging.
An Inconsistent Attempt
Then, in the course of looking for good pictures of the Parthenon, I came across my first Taj Mahal/Golden Ratio depiction. At first glance it looked like I had missed a genuine case of a golden rectangle in the Taj Mahal.
When I did a little more serious searching for the Taj Mahal and the golden ratio, I found the same picture that I had run across earlier. In fact, it appears all over the internet. I have no idea who first drew it up or how it got attached to a second picture, which one could take to be a close-up if one blurred the lines by squinting. However, the two pictures are not only inconsistent with each other, but neither of them works as a stand-alone example either. Apparently this set of two pictures is being copied from website to website, and people do not even look at what they're pasting.
Just so you know that I'm not just making this up I'll provide one reference, a site called "Project Steam," about which I know nothing more than that it has the commonly used set of pictures. Also, it is written in an engaging friendly manner. The author provides a catchy response to the idea that, since phi's decimals extend to infinity, it cannot be applied anywhere as a measurement. If I may quote,
Now, I am not a mathematician; I am an artist. So my reaction to this argument is to shrug and say, “Meh, close enough.” If that sort of blasé attitude offends your mathy sensibilities, you should probably stop reading now.
The math fan inside of me is taken aback; my writer's instinct loves the prose. I compromise and say that, if you're writing about math, you should avoid obvious goofs, particularly when they have nothing to do with math per se. My concern should be as clearly visible to the artist as to the recreational math fan, maybe even more so because they involve the placement of lines, not the calculations of equations. You can see the manipulations without knowing any math. As I said, I don't know who first came up with these pictures or who first decided to couple them. Still, I don't understand how anyone could post them without noticing the glitch.
I should mention that the writer's reaction to the criticism, though not the one that I would prefer, is on target. For all geometrical objects and the numbers associated with them, including not only irrational ones, but rational ones as well, we're always dealing with approximations. Would it be wrong for me to say that the wheels on my bicycle are round since pi is only an approximation? Of course not. Similarly, I'm also within my rights to claim that I have detected a golden ratio somewhere despite the fact that phi can never be more than an approximation. (Eventually, I will discuss the idea of the reality of numbers a little more thoroughly.)
Here are the two pictures in question, already combined into one image on that website so that the discrepancies should have been easy to catch.
Picture 1 (Internet Staple) Picture 2 (Internet Staple)
Picture 3 (mine) Picture 4 (mine)
The rectangle around the entrance does indeed come close enough to the value of phi to satisfy not merely the aspiring artist, but also the casual math player in me. But wait! There is an issue with how the rectangle is arranged, namely with the top line being above, i.e.,outside of, the light-colored frame, and the lines of the sides inside of it, as highlighted with the teal lines in Picture 4. That rectangle makes a hash of how the entrance is actually constructed and framed. I could accept the presence of a golden rectangle if it were all inside of the frame or all outside of it. But the way it is presented in Picture 1 is clearly not in line with the intentions of the artisans who built it.
And now, if I may, I would like to direct your attention to the close-up as shown in Picture 2 on the top right. In that picture the rectangle encloses the entire outside of that decorative frame. Did I get my wish? Sadly no. Even without measurement the difference in proportion between Picture 1 and Picture 2 should be evident. The difference strikes me as undeniable. My measurements came out to a ratio of 1 : 1.3, not close enough for anyone I would hope and certainly not what we see in Picture 1. How can one miss the difference in where the lines are drawn? Maybe the artistic author re-posted the pictures and didn't pay much attention because he didn't think it would make any difference. Presumably someone else came up with one or both of those pictures, and someone thought it would be a good idea to combine them, perhaps thinking that the inconsistency is just nitpicking or maybe not even noticing it. But it does matter, and accuracy is important. Phi is all about a number, and a different number can't substitute for phi. I don't intend to launch into a lecture on the virtue of precision in whatever you do. "Good enough" may very well be good enough at times. However, this is not one of those. If you say that you're illustrating the golden ratio, a very precise feature, and the illustrations are not illustrating the golden ratio, what good are you doing? The choice here seems to be between either an arbitrary golden rectangle that disregards the architecture and decorations or a rectangle that follows the features of the building, but not the golden ratio.
My point, once again, is simply that golden-ratio-mania is leading GREs to find phi all over the world, linking the beauty of a building to a specific number, and imposing it on objects where it is neither present nor needed. I'm not opposed to finding the golden ratio where it is, and if it's contributing to the beauty of something, very well. But if we feel as though we need to find phi in some contorted way in every beautiful structure, we are doing ourselves and the object a disfavor because then we are mechanizing our aesthetic sensibility.
I have had the privilege to visit the Taj Mahal. When you first walk onto the compound you cannot see it because your view is obstructed by a rather high wall, and you're too close to look over it. Then, once you walk through the inner gate, there it is, the Taj Mahal, right in front of you, larger than life--and incredibly beautiful. No reproduction that you've seen before can really do justice to the magnificent structure now in full view. Does it embody the golden ratio? I can't find it. Is it an unbelievably gorgeous sight? Absolutely. Would it be more beautiful if it did manifest the golden ratio? I don't see how it could be.
An Inaccurate Attempt
Once again I escape into the real world of real numbers, exploring the uses and misuses of phi. I mean, what we used to consider the “real” world has become so surreal at the moment that I can’t bring myself to write on it. Besides, there’s no shortage of commentaries on that realm. So I seek shelter in the part of the world that’s not going to change and its Creator. If you’re tired of reading about phi, or never were interested to begin with, I understand. But please don’t repeat the time-worn myths concerning phi after passing up this opportunity to reflect on the matter under the gentle guidance of your devoted bloggist.
In the previous entry, I focused on one particular set of pictures supposedly demonstrating how the golden ratio shows up in the entryway to the Taj Mahal, but that this case rests on a glaring mistake, which anyone should have been able to catch apart from knowing any math. Actually, if you examine a larger area than the entry gate, the reason for this strange placement of the golden rectangle entry becomes a little clearer, but no less arbitrary. GREs seem to be driven by an additional desire for bigger and better golden rectangles, and the inconsistency concerning the entryway not only gets copied from site to site, but is actually expanded by some further dubious interpolations.
I’m going back to the broadly circulated set of pictures that I labeled Pictures 1 and 2 last time. Here is Picture 1 by itself, and, in order to make it easier to talk about what’s happening here, I’ve labeled some important junctures of the lines with letters.
Among the various lines, two apparent golden rectangles are created. One can be described by ACFH, and the other one by BDEG.
These two rectangles can only be golden if they overlap, which they do. My measurements fall into the levels of tolerance that we cannot help but allow for. Each rectangle starts from the inside of the opposite decorative door frame and goes either left or right to the edge of the building as it is visible in a picture straight frontal picture. These extensions are supposed to be squares, and we know that a square added to a golden rectangle creates a new, larger golden rectangle.
In this illustration, the two squares are ABGH and CDEF—together with rectangle BCFG—are thought to make up two new golden rectangles.
The idea of two golden rectangles created by overlap is clever, though it would have required a lot of subtlety on the part of the architects of the Taj Mahal. I can’t say how such a construction would fit in with the supposed aesthetic appeal of phi. Is our vision supposed to shift back and forth, first catching this rectangle, then that one? For all that I know, such may be the theory and, given the initial assumptions, it could be true, I suppose. But it’s also a leap, and I'm not convinced of the assumptions. We’ve already found that the golden rectangle, as placed in in the entryway, compromises the light-colored decorative frame as a feature of the building (which is then covered up in the alleged close-up shot that I have called Picture 2 in the previous entry). I don’t know which idea came first, the two larger, overlapping golden rectangles or the imposition of the golden ratio on the entryway. Regardless, in either case, the arbitrary choice with regard to the doorway is still a hindrance.
Moreover, we also need to question the geometrical integrity of the two supposed squares. It should be immediately obvious that points A and D have no architectural anchorage whatsoever. They appear above the balustrade cutting through the small turrets at no particular locations of interest, except that they mark the end of the straight line outward from C to A and from B to D. The internet illustration make it just as clear as my depictions. Here is how it looks on one side:
The fact of the matter is that these squares simply do not exist. The front of the building ends before the square is finished and the wall is bent at an angle to form a new facet of the chamfered corner. (I just learned the word “chamfer.” It’s a decoration on what would otherwise be the stark edge of a 90° corner.) In the picture below you see how the top line of the supposed square ADFH does not stay on course heading left from B to I, though one can blame the photographic angle for that apparent anomaly. However, the shift at points I and J is integral to the building itself since the building has an edge there and begins a new facet. The decorations of the Taj include some optical illusions, but this is not one of them. There is no square here, but an edge in three-dimensional space, and I don’t think that it would occur to anyone looking at the Taj in real life, rather than as a flat picture, to see a square at these locations. I, for one, didn’t. Consequently, when combined with the contrived rectangle of the entry way, there is still no golden rectangle here. But, as I keep insisting, numbers are beautiful, but beauty does not depend on certain numbers.
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