Common Misconceptions about the Golden Section

[hamsterdance]

Note: Apologies if this has already been posted on WC elsewhere.
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Since trying to improve my compositions I've been doing some research on the subject - one of which included the Golden Section. As I did I started following up on mathematics websites instead of ones focusing more on the art aspects. In doing so I found a few refuting some of the claims made about past artists, paintings, architecture, music or whatnot that supposedly used the Golden Section.


One of the original sites reviewing a book on the Golden Section and misconceptions. The pasted sections didn't carry over the greek symbol so the sentences below are cut short or missing the subject sometimes.

You can also read the George Markowsky article as an Acrobat doc here.

Many readers of MAA Online may assume, since Professor Walser says that it has turned up in so many places since antiquity, that the golden section has been part of human consciousness for a long time. This is not so. The earliest use of "golden section" was in a German book published in 1835, a consequence, as one commentator put it, of "German romanticism". It did not appear in an English mathematical work until 1898. In 1509, Luca Pacioli called it the "divine proportion" but before that it was referred to, when it was referred to at all, as Euclid did, as "division in extreme and mean ratio." Pacioli's reasons for the proportion's divinity had nothing to do with geometry, architecture, music, or art. His reasons were all mystical and must be understood mystically. Roger Herz-Fischler gives them in his valuable A Mathematical History of Division in Extreme and Mean Ratio [1]. "The first is that it is one only and not more." "The second attribute is that of the Holy Trinity." The third was that .618... is irrational. The other reasons were equally mystical, that is to say irrational and not to be understood by using reason. Herz-Fischler says (page 150 of the Dover edition), "I insist upon the fact that Pacioli does not recommend the use of [the golden section] in determining the proportions of works of art and architecture." The belief that has, through the ages, been purposely built into buildings, paintings, or sonatas, or that it is somehow part of the world or of our brains is one that has sprung up fairly recently.

For example, the idea that a rectangle with dimensions and 1 (or, equivalently, 1 and - 1) is the one that is aesthetically most pleasing seems to have gotten a start in the 1860s, though too many authors repeat this as if it were part of the wisdom of the ages. (In what follows, I am drawing on another invaluable resource [2].) Back then, one Gustav Fechner presented subjects with ten rectangles and asked them which they thought was the nicest. The rectangles varied from a square to one whose sides had the ratio of 2 to 5 — that is, with aspect ratios from 1 to .4. The three rectangles in the middle, those with aspect ratios .57, .62, and .67, were chosen by 76% of the subjects.

Well, of course. Squares are dull, long flat rectangles look as if they had been stepped on, and tall skinny rectangles make us nervous-they look as if they may fall over any minute-so naturally something in the middle gets picked. But the golden section has nothing to do with it. Further studies have shown that -rectangles are in fact not the prettiest. There is now (at least there was when this was written) a Web poll on the subject, which may be found at http://homepage.esoterica.pt/~madureir, that has rectangles with width 54 and heights 65, 68, 72, 77, 81, 87, 96, 108. The sixth is close to a golden rectangle, but it has been picked as the most pleasing rectangle by only 12% of the 1501 respondents to the poll. The 54-by-72 rectangle is the clear winner, with 30% of the votes. (The percentages of respondents choosing each of the eight rectangles are, respectively, 9, 3, 30, 16, 19, 12, 5, 8.) If you measure books, a handy source of rectangles, you will find that almost none has dimensions that come close to those of a golden rectangle. A golden-rectangle book looks too tall and skinny. Professor Walser's book measures 9 inches by 6 inches, not 6 inches by 6 inches.

The facts are similar for the other bits of nonsense that are attached to [greek symbol]. Someone once said, perhaps on mystical grounds, that the height of the human body was divided at the navel in the ratio of 1 to , but it is not so, even for the most aesthetically pleasing of humans. People have gone to the trouble of measuring navel heights, one not being able to resist mentioning a "ticklish subject," and found that the ratio is, on the average, 1 to something larger than The ancient Egyptians did not use in designing their pyramids. How could they, since they had no notion of irrational numbers? The Parthenon does not have built into it, as many authors have asserted-the evidence seems rather better for 9/4 = 2.25. It must be the human thirst for marvels and wonders, combined with the human ability to see patterns where there are none, that accounts for such assertions.

Artists did not use [greek symbol] when composing, even though the writings of those whose heads have been turned by are full of pictures of pictures, from Michelangelo to Seurat and beyond, with golden rectangles in them. I would undertake to find a golden rectangle in any picture, even a Jackson Pollock: pictures have so many points of significance in them that would not be hard to find four that come close to forming a rectangle similar to a 1-by- one. Indeed, on the cover of Professor Walser's book we see a reproduction of da Vinci's Mona Lisa with three golden-looking rectangles superimposed on it. Two of them may have been inserted for decoration only, but the one framing the subject's face looks as if it was there to illustrate Leonardo's use of . But it doesn't quite fit. Its right-hand side almost goes to the start of the hair but not all the way; its lower edge is a bit below the chin. Did Leonardo lay out a golden rectangle before starting to paint? I suppose we cannot know for sure, but I think that the face came first, not any geometry.

....Click on link above to finish the article.

And another...

There are many books and articles that say that the golden rectangle is the most pleasing shape and point out how it was used in the shapes of famous buildings, in the structure of some music and in the design of some famous works of art. Indeed, people such as Corbusier and Bartók have deliberately and consciously used the golden section in their designs.

However, the "most pleasing shape" idea is open to criticism. The golden section as a concept was studied by the Greek geometers several hundred years before Christ, as mentioned on earlier pages at this site, But the concept of it as a pleasing or beautiful shape only originated in the late 1800's and does not seem to have any written texts (ancient Greek, Egyptian or Babylonian) as supporting hard evidence.

At best, the golden section used in design is just one of several possible "theory of design" methods which help people structure what they are creating. At worst, some people have tried to elevate the golden section beyond what we can verify scientifically. Did the ancient Egyptians really use it as the main "number" for the shapes of the Pyramids? We do not know. Usually the shapes of such buildings are not truly square and perhaps, as with the pyramids and the Parthenon, parts of the buildings have been eroded or fallen into ruin and so we do not know what the original lengths were. Indeed, if you look at where I have drawn the lines on the Parthenon picture above, you can see that they can hardly be called precise so any measurements quoted by authors are fairly rough!

So this page has lots of speculative material on it and would make a good Project for a Science Fair perhaps, investigating if the golden section does account for some major design features in important works of art, whether architecture, paintings, sculpture, music or poetry. It's over to you on this one!

Article: George Markowsky's Misconceptions about the Golden ratio
in The College Mathematics Journal Vol 23, January 1992, pages 2-19 is an important article that points out the weaknesses in parts of "the golden-section is the most pleasing shape" theory.

This is readable and well presented. Perhaps too many people just take the (unsupportable?) remarks of others and incorporate them in their works? You may or may not agree with all that Markowsky says, but this is a good article which tries to debunk a simplistic and unscientific "cult" status being attached to Phi, seeing it where it really is not! This is not to deny that Phi certainly is genuinely present in much of botany and the mathematical reasons for this are explained on earlier pages at this site.

Article: How to Find the "Golden Number" without really trying
Roger Fischler, Fibonacci Quarterly, 1981, Vol 19, pages 406 - 410.
Another important paper that points out how taking measurements and averaging them will almost always produce an average near Phi. Case studies are data about the Great Pyramid of Cheops and the various theories propounded to explain its dimensions, the golden section in architecture, its use by Le Corbusier and Seurat and in the visual arts. He concludes that several of the works that purport to show Phi was used are, in fact, fallacious and "without any foundation whatever".

Article: The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh Col. R S Beard in Fibonacci Quarterly vol 6, 1968, pages 85 - 87;
has three separate theories (only one of which involves the golden section) which agree quite well with the dimensions as measured in 1880.

Article: Golden Section(ism): From mathematics to the theory of art and musicology, Part 1, Dénes Nagy in Symmetry, Culture and Science, volume 7, number 4, 1996, pages 337-448
Section 2.1 says there are at least nine different theories about the shape of the Great Pyramid of Pharoah Khufu (the Great Pyramid of Cheops), two of which refer to the golden section:
The angle of the slope of the faces is

* the angle whose cosine is 0·618... which is about 51·82°
* the angle whose tangent is twice 0·618... which is about 51·027°

although a better fit is provided by a mathematical problem in the Rhind Papyrus which, in our notation is

* the angle whose tangent is 28/22 which is about 51·84°

Edits: Changed order of articles. I like the first one listed now better than the second one quoted.

[vhere]

Interesting what a short history it has

'Well, of course. Squares are dull, long flat rectangles look as if they had been stepped on, and tall skinny rectangles make us nervous-they look as if they may fall over any minute-so naturally something in the middle gets picked'.



I paint on squares and long thin tall rectangles a lot - because I think conventional rectangles look dull and .... well .... very conventional. Squares and long thin canvasses have more exciting dynamics and drama - for me anyway

each to their own!

the golden mean is interesting in the way that it creates spirals and in that way occurs in nature - but as an unbreakable rule for picture making creates some very dull paintings.

The shape of the painting should suit the subject and if the subject needs a traditional rectangle - that's fine. Lots of images are much better with a variety of other shapes.

Vivien

[Celeste McCall]

This makes me wonder why facial features aren't divided at .5 vertically.


And why our fingers and toes and legs, etc. aren't divided at .5. Yet, the body, if folded down the middle....is equally divided.

Additionally, I wonder why there can be so many ways to make an apple pie, and, yet, all of them be really tasty. And, then again, without the apples or the pastry....it wouldn't be apple pie.

Hummmmmmm......

Good articles.

Thanks for sharing them.

[eLBeJay]

this topic, also know and the gloden ratio, the golden mean, the sacred ration, ect. is very addicting, and can at some points suck u in, so beware. it happened to many artist in the past. sssooo...here is a refrence on how to get sucked in deeper. check out a book called "zero, the biography of a dangerous idea. it will send u into a wirl wind of mathimatical consepts to throw on a canvas or two...maybe even three or more. have fun.

[deadly_ambition]

Have you read the painter's secret geometry? They took apart major paintings and showed how the artists used phi to conpose their pictures. It seems pretty convincing.

[vhere]

Originally Posted by deadly_ambition

Have you read the painter's secret geometry? They took apart major paintings and showed how the artists used phi to conpose their pictures. It seems pretty convincing.

mmm art historians can argue all sorts - did the artist actually ever intend half of what they claim though?

I certainly wouldn't want to create a painting based on maths and geometry. I think it is often pure chance rather like statistics, there are several ways of looking. Lies, d*** lies and statistics/golden mean?

[LarrySeiler]

Originally Posted by vhere

I certainly wouldn't want to create a painting based on maths and geometry. I think it is often pure chance rather like statistics, there are several ways of looking. Lies, d*** lies and statistics/golden mean?

art ed will often show that the left brain is a bit like the older insistent sibbling that wants control over everything and everyone...and that people often operate, or most operate with the voices of the left brain hounding them. That voice is mindful often of the time it is, where one needs to be, how things need to look, who might be looking, do this and do that, with plenty of judgment to fuel its cause.

The right brain is a bit more like the sloppy careless individual, often late, throwing the jacket on the back of the chair, carefree, irresponsible as far as the left brained individual might be.

The left brain tends to dominate some individuals more....the right brain others. It is understood that the right brain is the hemisphere that responds more to shape, color...creativity and so forth but for some its held too greatly suspect where their left brain dominates.

That is not to say left brained persons can't create art...but it would not be surprising to see step by step analytical treatise, caution, and interest in assurances to guarantee success.

The Renaissance was a great time not only of the birthing of humanism really, but in believing that if there was a God...He was a God of order and design. Such order can be understood (analytical) and understood, such ordering can be applied to creating/making art.The problem with this ordering approach is its a bit like the right brained child attempting to do homework with the left brained older brother standing behind looking over the shoulder ready to smack him upside the head at every turn.

I have Da Vinci's Anatomy of Trees...which is an inciteful interesting book but what a bore to take serious as an artist and try to work thru and apply. Filled with math and logic to come to a determination the dimensions of a tree's trunk, to its limbs, the branches.

It is a left brained approach entirely whose aim is not only in understanding but to assure success.

The artist has a couple courses of action to go...one will use an approach that would be restraining and stifling to some, another might go with what will seem careless..haphazard and irresponsible to others.

As it turns out...my paintings might appear to many to apply to the golden means because my focal points of interest happen to be found there...however I intuit such things from years of experience and painting. It may be reckless and careless...too carefree, but my heart has freedom with no burden of it.

As an art instructor...knowing the fear some have and the need to feel assured, I will teach and refer to the existence of this method.

Here's where I'm coming from though...I think of it like romance.

There might not from appearance appear to be much difference between two gents courting and romancing their love...but one might have their whole heart and attention on the girl of their dreams while the other is plotting and working each thing out in their mind. "what should I do next...what shall I say?"

How creepy do you think it would be to this young woman to know that in the mind of her lover are many readings and advice, many step by steps going on depending on her moment and receptivity?

I want my painting to be a product of my absorption and passion for my subject and I've learned to trust my aesthetic instincts which have proven again and again to be reliable.

Instead of measuring tree trunks, I'll squint my eyes...see the language of art known and spoken by artists such as shapes, color, values, texture, form, contours and so forth. I'll block in masses in general color, appropriate value...use negative space to sculpt the form to take on the character of what I'm seeing and I'll leave it to the viewer to see the resulting illusion and think "tree!"

I'm not criticizing the analytical approach if some need it....but if one can ride the right brain wave, why fix then what is not broken?

Larry

[vhere]

Originally Posted by LarrySeiler

I'm not criticizing the analytical approach if some need it....but if one can ride the right brain wave, why fix then what is not broken?

Larry

I agree totally and that last sentence sums it all up neatly

[AndyRichardson]

Sooo, Hamsterdance, what do you think now, about the notion of the Golden Section? Has the study helped any?
I know you said that it was done in order to improve on your compositions. And, I suppose like all of us here, you wanted to research the sources of the "rules" of composition, perhaps with the intent to free yourself of mindlessly and slavishly having to follow the rules; perhaps to see what latitude you might have to vary from such strictures? Nothing wrong with such an effort, and I wouldn't characterize it as a left-brain quest if its intent was to think out-of-the-box.
Another area of study that might prove to free your artistic hand (by delimiting or bracketing the boundaries of what is considered established fact) would be to check out some of Kandinsky's comments about the elements of composition. He directly spoke to the feelings associated with the elements, much as you evidently found in:
'Well, of course. Squares are dull, long flat rectangles look as if they had been stepped on, and tall skinny rectangles make us nervous-they look as if they may fall over any minute-so naturally something in the middle gets picked'. In Kandinsky's case, he had a brain arrangement that allowed him to hear sounds whenever he sensed certain colors and forms (synaesthesia), so he was more tuned in (literally) to his compositional form intuitions. But, since his brain worked differently, his paintings are usually received as "yuck" by us normal perceivers.
Hans Hofman was another one who would be an interesting read, especially about the sense of depth in abstract forms.
Then, after spending about ten years absorbing and intellectualizing and freeing yourself, you can then get around to actually painting!
That little dig being said, I must admit that it is fun to creatively explore the penumbra surrounding artistic production. Such an effort can be very intuitively 'compositional' and creatively satisfying in itself. Too bad we can't all share the compositions, the arrangements-of-our-derangements in our minds, as easily as our pictures allow us to do!

[FriendCarol]

Painter's Secret Geometry is by Bouleau, I believe, and was first published in 1951. The notions in it were rediscovered over the past decade, independently, by the author of handprint.com -- see his page "Format Proportions."

The 'golden mean' or whatever you want to call it, definitely exists in nature. I was taught about it in some middle school art class with regard to classical paintings. I very much doubt that a value known to ancient Greek philosophers only surfaced again in the 19th century! Have we forgotten that the Renaissance was brought about, in part, because old manuscripts and forgotten lore was rediscovered?

As to the measurement of humans, we've been growing, on average, for some centuries. Go look at some medieval coats of arms, if you don't believe me. I don't know how this (surely a result of improved nutrition on average, in developed countries) has affected our proportions, but we apparently use a different number of head-lengths now for total height, in figure drawing.

Then, after spending about ten years absorbing and intellectualizing and freeing yourself, you can then get around to actually painting!

Some of us like to think AND paint. We can even do more than one of these in the same day.