Every one knew how laborious the usual Method is of attaining to Arts and Sciences; whereas by his Contrivance, the most ignorant Person at a reasonable Charge, and with a little bodily Labour, may write Books in Philosophy, Poetry, Politicks, Law, Mathematicks and Theology, without the least Assistance from Genius or Study.

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Laputan Logic

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Laputan Logic*
Fanciful. Preposterous. Absurd.
The Cult of the Golden Ratio

Posted on Friday 19 January 2007

What role did the Golden Ratio play during those terrible events in November, 1963?

Honestly, people spout a lot of crap about the Golden Ratio .

I mean that, to paraphrase the Willard character out of Apocalypse Now , there's just so much bullshit piling up on this subject that you practically need wings to stay above it. The Golden Ratio, once a pristine jewel of geometrical truth and simplicity, has become a deity for a cult of hyperlinking headnodders whose chief devotional practice seems to be to handwave their way from one disconnected and unexamined falsehood to another.

Like many cults, the founding myth of this one is the belief that its truths are very old - that this is the occult wisdom of the ancients. In fact, it is actually quite modern and has its roots not with, say, Pythagoras or ancient Egypt but rather with the restless young men of German Romanticism.

While the ancients certainly knew and understood about the Golden Ratio, they didn't invest it with any special significance apart from its obvious geometrical utility. Some Renaissance thinkers, on the other hand did see a mystical significance in the number (which they dubbed the "divine proportion") but this fancy was of a very different nature to the modern version. For a start, these were mathematicians with a strongly Christian Platonist worldview. The Golden Ratio was divine because of its importance in the construction of the Platonic solids and other polyhedra .

I see Golden Rectangles. Count them, there are two.

Modern devotees of the Golden Ratio by comparison are in general rather feeble at geometry. Their main interests lie in divining Golden Rectangles (i.e. rectangles where the ratio of the sides are equal to the Golden Ratio) from photographs of churches and paintings. Golden Rectangles, they argue, have always been appreciated as the most aesthetically pleasing of all rectangles and this is why they were used extensively in the architecture of classical temples, most notably in the Parthenon in Athens.

The reality is however that there is not a shred of evidence to support this claim. When actually measured the Parthenon does not exhibit any definitive Golden Ratios. In fact after close examination it becomes clear that the temple's construction is not very precise with things such as column height and spacing varying all over the place. Apart from this, there is good reason to think that the Greeks probably didn't have used the Golden Ratio in their architecture. For one thing, the number is irrational (in fact it is the most irrational of all irrational numbers ) something that would have made it extremely difficult to work with using their system of whole numbers and ratios. Vitruvius, the only ancient author whose work on architecture has survived, briefly mentions that the architect of the Parthenon, Ictinos, actually wrote a book which explained its proportions but unfortunately he doesn't elaborate any further. Despite this knowledge, however, Vitruvius makes absolutely no reference to the Golden Ratio anywhere in his work, staying instead with simpler rules of proportion.

The logarithmic spiral shell of the Nautilus pompilius

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 (one might imagine that being a totem and being collected to extinction are two not necessarily unrelated phenomena).

Certainly, there is no question that the nautilus spiral is mathematically interesting. D'arcy Thompson pointed out in his 1917 classic On Growth and Form , that the shell grows in the shape of a logarithmic or equiangular spiral and this enables the creature inside to grow steadily without needing to change its shape. Also a recent mathematical study was made to measure the fractal dimension of these shells .

The logarithmic spiral is certainly a nifty curve in its own right, so much so that Jakob Bernoulli dubbed them Spiral Miribilis and even had one engraved on his tombstone (alas it turned out as an Archimedian spiral). But w hile 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.

Logarithmic spiral based on the Golden Ratio. The spiral of the nautilus shell.

The claim that the Golden Rectangle is the most "pleasing " comes to us 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 naval 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.

Georges Seurat , The Parade, 1899

After Zeising's time the enthusiasm for finding Golden Rectangles has only grown in art and architecture. The first artist to have his paintings analysed for Golden Rectangles was the French pointillist, Georges Seurat. These conjectures ultimately proved to be groundless.

Attention has since shifted to the works of Leonardo Da Vinci whose gigantic reputation as an innovator and visionary, one imagines, would have been too attractive to resist. Leonardo's connection with the the Golden Ratio is strengthened by his association with Luca Pacioli, a Franciscan friar and famous mathematician (and patron saint of chartered accountants) who wrote a book on the ratio which he entitled The Divine Proportion . Leonardo himself provided the illustrations for the book.

Leonardo may indeed have explored the use of Golden Ratio, perhaps due to Pacioli's influence, however there is absolutely no evidence that he did. Instead we have evidence that he used approximations to draw his pentagons, not something one would expect from a person supposedly enamoured with the Golden Ratio. Where some people have proposed that he used it in the composition of his paintings, none of the suggested places where he is said to have used it (such as in the dimensions of the face of the Mona Lisa or the body of the unfinished St. Jerome) are particularly convincing. Of the countless rectangles proposed none fit very well and this lack of precision undermines the basis of any argument that is based on geometry. In addition, it needs to be noted that the Virgin on the Rocks , St. Jerome and the Last Supper where all painted by Leonardo some years before he even met Pacioli.

Now please don't get me wrong. The Golden Ratio really is a remarkable number which many interesting mathematical properties – some of which I hope to explore sometime in a future post. But for me what I find so astonishing and frustrating about it is the role that it serves as magnet for so much nonsense, something that it seems to attract as inexorably as a belly button attracts lint.

La Gioconda.Today we decide to measure the face from the hairline rather than the top of the head. (Not that anyone will notice)