Let’s not bother with facts and definitions as we all know what we’re talking about here. However – and I’ll return to this – the A-series paper sizes are not trying to look beautiful.
Here’s one of many websites devoted to the Golden Proportion. I’m not the first to blog about the cult or myth of the Golden Ratio. This post, over at Laputan Logic, is one of the best. Amongst other things, it points out that this is not a logarithmic spiral.
A creature which has been greatly misused by Golden Ratio cultists is the poor old Nautilus pompilius. As if being pushed to near extinction wasn’t bad enough, the beautiful spiral shell of this animal, which is a relative of the octopus, has become a sort of totem for graphic designers who never fail to resort to it whenever they need a graphic to grace an article or book cover that might even tangentially refer to the Golden Ratio. But while it is rare to find an article featuring the Golden Ratio that doesn’t feature a luscious image of one of these shells, the reality is that there is no real connection between them. There are certainly many ways of parameterising a logarithmic spiral so as to closely match the curve of a nautilus shell but none of these except to most contrived comes anywhere near to the Golden Ratio.
Lap also gives suggestions for further serious reading. It’s an old site but this link is still live and a good one. Nobody is denying the fact that the Fibonacci Sequence exists in Nature and is present in the unfurling of petals, the packing of seeds and the distribution of leaves.
The amazing thing is that a single fixed angle can produce the optimal design no matter how big the plant grows. So, once an angle is fixed for a leaf, say, that leaf will least obscure the leaves below and be least obscured by any future leaves above it. Similarly, once a seed is positioned on a seedhead, the seed continues out in a straight line pushed out by other new seeds, but retaining the original angle on the seedhead. No matter how large the seedhead, the seeds will always be packed uniformly on the seedhead. [Thanks mathgeek]
That’s all fine. But then people started to find and see the Golden Proportion everywhere.
The big mistake is to assume that because things that might please us in the natural world use the Golden Mean, applying it to artificial things like paintings and buildings will therefore make them beautiful as well. This is not logic. It is an attempt to generate a notion of beauty by association.
These last two images, you’ll notice, find the Golden Proportion in completely different places. The following quote I’ve simply lifted from Laputianlogic.
The claim that the Golden Rectangle is the most pleasing comes to us via Adolf Zeising who is the one who single-handedly started the whole Golden Ratio craze in the first place. In 1855, he published a book which he modestly entitled: “A New Theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole nature and art, accompanied by a complete summary of the prevailing systems.”
It was from him that we learn that the proportions of the human body are based on the Golden Ratio. For example, taking the height from a person’s navel to their toes and dividing it by the person’s total height yields the Golden Ratio. So, apparently, does dividing height of the face by its width. From here Zeising made the connection between these human-centred proportions and ancient and Renaissance architecture.
Not such an unreasonable jump, to be fair, but the connection to the Golden Ratio had no basis in reality. When measuring anything as complex as the human body, it’s easy to come up with examples of ratios that are very near to 1.6 (or 5/3). But there’s no need to jump from here to any conclusions about the Golden Ratio. [Laputianlogic]
Neufert (1900–1986), a disciple and employee of Walter Gropius, combines rational norming with an aesthetic impetus. He propagates the Golden Ratio as this architectural principle of proportion, that together with his own normed measures leads to a “spiritual permeation” and a renewal of architectural formation by “an inner law” in the spirit of Antique, Gothic, Renaissance, and Classicism of Palladio and Schinkel [Neufert 1936: 30]. [The Golden Section in Architectural Theory]
In The Golden Section in Architectural Theory, Marcus Frings debunks the whole idea of the Golden Proportion as a generator of Renaissance art and architecture. Nevertheless, the Golden Section travelled around the world, starting with Palladio
and reaching England via Inigo Jones. It’s application and associated stylistic hijinks quickly found favour with the upper classes. By the time the style trickled down to the lower classes of Georgian housing, only the window proportions and spacing remained. This process is merely the adoption in lower-class housing, of simplified (aka less expensive) features of the upper class housing at the time – aspirational decoration, in other words, and all achieved by only the proportions and spacing of windows. Elegant simplicity. Cheap and cheerful. Genius! The popularity of the Georgian townhouse in part rests on it being an aspirational product which doesn’t cost anything extra to build. Over time however, what we have is the continuation of the 2,500 year link between the Golden Proportion and the architecture of an elite.
It was Zeising who first made the connection between Classicism and Nature. If one combines the ancient Greek affinity for the Golden Section along with its practical applications in Nature (as opposed to possible sightings) and then add a bit of marketing savvy, then we have the perfect conditions for the marketing of architecture. Wright had two out of the three but Corbusier put it all together first.
By peppering one’s elevations with the Golden Section, it’s possible to be Classical and Romantic/Natural/Organic at the same time. It’s a powerful thing, especially when you’re trying to sneak in new building economies (such as the absence of ornament) under the table.
Colin Rowe’s 1947 essay The Mathematics of the Ideal Villa perpetuates the myth of the Golden Proportion and uses it to explicity link Classical architecture, Corbusier and “natural” beauty.
Thus, either because, or in spite of theory, both architects share a common standard, a mathematical one, defined by Wren as “natural beauty”; and within the limitations of a particular programme, it is not surprising that the blocks should be of corresponding volume – 8 : 5½ : 5. Corbusier has carefully indicated his relationships by regulating lines, dimensions and figures, and over all he places the ratio of the golden section, A : B = B : (A+B). Thus he indicates the ideal with which he would wish his façade to correspond, although in actual fact the figures 3 : 5 = 5 : 8 thus represented are only approximate.
But how natural is all this? The application of the Golden Mean to building plans or façades whether Corbusian, Palladian or Athenian is an example of biomimicry but with no functional advantages to be gained. It is clear that we have a preference for The Golden Section but what we don’t know if it is any of the following.
One. Because we are still keen to ape the Greeks? No.
Various authors discern golden ratio proportions in Egyptian, Sumerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from the late Bronze Age, which predates by about 1,000 years the Greek mathematicians who were first known to have studied the Golden Ratio. However, the historical sources are obscure, and the analyses are difficult to compare because they employ differing methods. [w]
Two. Maybe we are simply educated to appreciate the Golden Proportion and use it as shorthand for “pleasing” or “beautiful”, however meaningless. In other words, it’s a cultural thing. Amongst architects, I suspect this is the case. This is Frings’ conclusion.
For a long time the Golden Section does not occur in architectural theory. It first appears in the 19th century, through Zeising and Fechner, and then rises to a certain fashion in the third and fourth decade of the 20th century, from which Neufert and Le Corbusier get to know it. Neufert held out great hopes for a renewal of architecture through the Golden Mean, but he soon became sober.
After early experiments Le Corbusier uses the Golden Section to develop his catalogue of measures, which has — due to roundings and combinations — not much in common either with the Golden Mean or with the Fibonacci series. In fact, Neufert and Le Corbusier seem to use the Golden Section as a way to embellish their own subjective artistic creation by theory and ratio. In any case, the Golden Section certainly does play a role in the writings of these architectural theorists. Prior to the 19th century, however, the Golden Section is simply absent in written architectural theory.
Three. Maybe it’s because our eyes, due to their physiology, actually have a preference for this proportion. In other words, they can’t help it, they’re made that way. This seems increasingly likely. I was looking for some more information on things like the position of the optic nerve vis-á-vis the retina/cornea or something and came across this interesting link about chemical and biological relationships. Treat it with suspicion as it’s very keen to use the term Divine Proportion. This site however, contains some useful-looking information about the relationship between hydrogen bond distances and diffraction patterns in quasicrystals but it only leads to further questions such as “is the cornea a quasicrystal?”. Answer: YES.
Quasicrystals is exciting new stuff. The Nobel Prize Committee thinks so.
2011 Nobel Prize in Chemistry: ‘Quasicrystals’ Once Thought Impossible Have Changed Understanding of Solid Matter
Oct. 5, 2011 — The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Chemistry for 2011 to Daniel Shechtman of the Technion – Israel Institute of Technology in Haifa, Israel, for the discovery of quasicrystals: non-repeating regular patterns of atoms that were once thought to be impossible.
Just in passing, here’s a link to a site that explores computer-generated two- and three-dimensional Islamic Star patterns.
Moving on though, this next quote, from soicionomics clarifies the link between the Golden Ratio and all biological structures. Sensibly, it refers to the Golden Whatsit as φ (phi). This is probably what I should have been doing all along.
The source of all these biological structures is DNA. Given current best measurements, the length of one DNA cycle is 34 angstroms, and its height is 20 angstroms, very nearly producing the Fibonacci ratio (see Figure 15). Stanley et al. note parenthetically in their power-law study, “The DNA walk representation for the rat embryonic skeletal myosin heavy chain gene [has a long range correlation of] 0.63,”16 which again although not mentioned in the study is quite close to phi. A bit of data integration, then, shows that living systems are permeated with phi-based structures.
We can be pretty sure that
- DNA is not trying to look beautiful
- the form of DNA has something to do with the functioning of DNA.
All organic matter contains φ. If diffraction patterns in the quasi-crystals that make up our retinas cause φ-patterns to somehow resonate or stimulate the retina more than non-φ patterns do, then what we have is a physiological explanation for certain notions of visual beauty.
This wikipedia entry introduces the work of Adrian Behan in Constructal Theory.
The constructal law is a first principle of physics that accounts for all design and evolution in nature. It holds that shape and structure arise to facilitate flow. The designs that happen spontaneously in nature reflect this tendency: they allow entities to flow more easily – to measurably move more current farther and faster for less unit of useful energy consumed. Rain drops, for example, coalesce and move together, generating rivulets, streams and the mighty river basins of the world because this design allows them to move more easily.
I mention Constructal Theory because it provides a greater scientific context for nature’s efficiencies such as φ. If the processes of nature can generate systems that accomplish specific goals with the minimum of resources and energy then, rather than imitate certain structures or systmes, it might be a good idea to apply this one and only principle into everything we design or make.
Four. The next and last question is “Is there is some evolutionary reason why our eyes should have developed so? What evolutionary advantage could there be to having eyes that respond preferentially to certain patterns and structures?” I’m no evolutionary biologist but if something to do with vision has an evolutionary or biological basis, then it’s probably because it enhances the ability to judge whether you can eat it, or whether it can eat you. Being able to quickly recognise living things would make for more efficient hunting and foraging.
To finish, the following is from sciencedaily.
Bejan argues that the world — whether it is a human looking at a painting or a gazelle on the open plain scanning the horizon — is basically oriented on the horizontal. For the gazelle, danger primarily comes from the sides or from behind, not from above or below, so their scope of vision evolved to go side-to-side. As vision developed, he argues, the animals got “smarter” by seeing better and moving faster and more safely.
“As animals developed organs for vision, they minimized the danger from ahead and the sides,” Bejan said. “This has made the overall flow of animals on earth safer and more efficient. The flow of animal mass develops for itself flow channels that are efficient and conducive to survival — straighter, with fewer obstacles and predators.”
For Bejan, vision and cognition evolved together and are one and the same design as locomotion. The increased efficiency of information flowing from the world through the eyes to the brain corresponds with the transmission of this information through the branching architecture of nerves and the brain.
“Cognition is the name of the constructal evolution of the brain’s architecture, every minute and every moment,” Bejan said. “This is the phenomenon of thinking, knowing, and then thinking again more efficiently. Getting smarter is the constructal law in action.”
While the golden ratio provided a conceptual entryway into this view of nature’s design, Bejan sees something even broader.
“It is the oneness of vision, cognition and locomotion as the design of the movement of all animals on earth,” he said. “The phenomenon of the golden ratio contributes to this understanding the idea that pattern and diversity coexist as integral and necessary features of the evolutionary design of nature.”
In numerous papers and books over past decade, Bejan has demonstrated that the constructal law (www.constructal.org) predicts a wide range of flow system designs seen in nature, from biology and geophysics to social dynamics and technology evolution.
Constructural Theory has huge applications for the design of buildings as well as their construction, the project management of that construction, and how we live in those buildings after their construction. The principle of economy of means is the one principle of Nature that we should be applying. Unfortunately, the overriding principle of high-end architecture is the opposite – decadence of means, and often in the form of using the maximum possible resources to achieve the appearance of a natural object that uses the least. Although Architecture has survived as a social phenomenon on this basis, it does not seem like a responsible way to head into the future.
The A-series paper sizes is one of the best-known examples of the Golden Proportion. It is also one of the few (only?) man-made applications of the Golden Ratio that comes anything near to what Nature achieves with its applications of the Golden Ratio. A-series paper sizes were first proposed in 1786 by the German scientist Georg Christoph Lichtenberg.
Georg Christoph Lichtenberg
for producing one of the few examples of humanity using the Golden Ratio
to create something with fewer processes and with less wastage
Misfits salutes you!