For over 2,500 years now we have been obsessed with a simple, if unlikely mathematical constant (1.61803398874989) which, from ancient Greece right up to LeCorbusier and beyond, has been arbitrarily chosen as the most aesthetically pleasing comparative height and width ratio for buildings (and other objects). Modern architects depart from it frequently, but always with an air of rebelliousness, indeed almost self consciously, therefore simply tacitly affirming its ever-present influence by so studiously avoiding it (and nowadays a swing in fashion appears to have restored its popularity in any case). Modern postcard manufacturers depart from it at their peril, and that has been proven in sales returns - the ultimate arbiter of general, if not lowest common denominator, public taste.

But what is it? And where did it come from? We know the Parthenon as a famous example of its use, but then one could almost pick any Greek or Roman temple and find its use. It is to be found in Palladian architecture, in Renaissance architecture, and even - it has been claimed - in traditional Japanese architecture amongst other non-western cultures (see below), thereby belying its origin in Greece alone. But then the Greeks simply formulated it in their geometry. They never claimed to have invented it anyway, choosing instead to regard it as so natural a pleasing formation to the eye that it had divine connotations. The 15th century monk Luca Pacioli would only have quibbled with Euclid over which god should be given the credit. It was he who rechristened it "the Divine Proportion", a name which has also stuck.

There is no doubt that its use is popular and has been for millennia. But is it a case of we like because we're used to it now, or that we're used to it now because it's innately the most likeable ratio to the eye? And if it is, then why is it so?

Doesn't the Golden Ratio turn up in Fibonacci sequences and is used in fractals? These appear in nature and that might suggest that it has some fundamental property that we respond to in an instinctive fashion.
Have we any mathematicians amongst us?

Which then begs the question, even if it's an instinctive reaction why is it an instinctively positive one?

The later Egyptian dynasties also seemed to ignore it in their buildings. Were they, like many modern architects, pointedly cocking a snoot at it or had they just poor instincts?

Oh heaven's, talk about being out of one's depth! I should stop this stream of consciousness posting, it's getting me into all sorts of bother.

However, I could possibly hypothesise that the occurrence of the ratio in natural forms has familiarised us to it and made it feel in some way right, an original and pre existing order, but that's utterly unprovable.
To go really beyond any sensible speculation, the Egyptians could have been promoting an ideology of being above and beyond the limits and constraints of nature and expressing their domination of it by deliberately employing another set of proportions.

I must stop making soup with the funny mushrooms!

Some have tried to speculate in antiquity that the perfect human form also adhered to the rule, though Da Vinci famously experimented with this and ended up with his "Golden Rectangle" instead, which isn't quite the same thing. There are some grounds to believe however that he used the Golden Ratio to compose the basic elements in some of his paintings. Long before Dan Brown it was well known that The Last Supper contained geometric consistencies which conform to Phi, though I dare say it was an intellectual puzzle he set himself rather than an attempt to tell Tom Hanks that Jesus was alive and well, looking hot and speaking bad English in southern France.

Oh heavens, as interesting as this is, I am way out of my depth too. But in art classes way back when Adam was a pup, I do vaguely remember something about Da Vinci and the golden ratio.

But, as you say Nordmann, Da Vinci wasn't using a secret code. He would have used the ratio as a guide for the composition of The Last Supper and the figures in it. Most artists used geometry when initially sketching out their work, triangles or arcs etc were also used in composition. Not only as a guide to assembling figures and objects in a pleasing and accruate way on the canvas but also to draw the eye of the viewer to the central theme or figure in the painting.

Actually, I've just noticed - if you look closely at the icon I've used for the "Civilisation" category on the main page the little Assyrian bullman is being (literally) lined up for Ratio compliance. Hadn't noticed that when I borrowed the image.

Wonder if he passed?

But that's really what my original question was getting at; if this proportion is the result of some intrinsic human aesthetic how far back can we trace it? When it comes to buildings, for example, how was this demand met before the technology of stone-cutting, brick-making and so on allowed an accurate rendition of its form? Were we as a species completely aesthetically frustrated for our first few hundred thousand years or did we scratch the itch some other way?

Nordmann,

we learned it many years ago as the "gulden snede" (the golden section) and as a over b is equal to a + b over b...

Found a complete debunking of the story called the myth of the golden section...

http://www.nlpvf.nl/book/book2.php?Book=46

Comments and synopsis of the content in Dutch:

http://www.nvvw.nl/page.php?id=7453

1835 first coining of the term "golden section" in the work of a certain Ohm. His brother we know better from our course electricity...

Some 20 years later a certain Adolf Zeising starts with the myth of the golden ratio relating it to nature and a "striving for beauty"...

And what says our first line help Wiki:

http://en.wikipedia.org/wiki/Golden_ratio

Yes and when even erudite academici are lurred into...who are we to...?

Kind regards and with esteem,

Paul.

Just thought you'd like this. THe Golden Ratio, and Fibonacci spiral have been found in Hokusai's 'The Great Wave off Kanagawa'. This should be of no surprise given the development of Wasan Geometry in Tokugawa Japan.

This is from my son, who is an architect (or hopefully will be after this weekend when he has a three-hour "conversation" with whoever judges these competencies). But he's not a historian. 　

　 　you're right. Don't have the historical background further than the Greeks. Does turn up in Fibonacci sequence, because it is uses the ratio (as said (a+b)/b, that is 3/2, 5/3, 8/5, 13/8, 21/13 etc getting closer to the Golden Ratio all the time, but works with any two starting numbers).　Would hazard a guess that the development of the concept of the proportion (and its harmony) was developed in parallel with the philosophy of mathematics and geometry, so wouldn't have been conceived of much earlier. Yes, was taught in architectural history in 1st year uni, and yes would say Nordmann's first post was spot in that it is very much still consciously there. Would only qualify that by saying it isn't the most economic ratio, at least when using 2.4x1.2 sheets of cladding, or 600x600mm tiles for instance, and that is normally much more important for clients.

Out of my field, but would guess its basis might be in our field of vision, and very much suspect different ratios would equally find pleasing forms and comparisons within nature if people were to look for them. Maybe the challenge is to find a really unattractive building using the ratio, or a really attractive one built entirely from squares.　

Falling-water - Frank Lloyd Wright attached. My favourite building. I don't see it...　

http://www.wright-house.com/frank-lloyd-wright/fallingwater-pictures/large-fallingwater-photos/high-resolution/25SW-falling-water-path2-L.jpg

My son had a photo of falling water, and some golden ratio lines across it, which he didn't seem to think matched this building. At least I think that's what the last sentence means.

nordmann wrote:

When it comes to buildings, for example, how was this demand met before the technology of stone-cutting, brick-making and so on allowed an accurate rendition of its form? Were we as a species completely aesthetically frustrated for our first few hundred thousand years or did we scratch the itch some other way?

In terms of the natural universe I had previously vaguely held that the lines of tropical latitude and/or the arctic/antarctic circles corresponded to the 'golden section' of the Earth's respective hemispheres. This seems, however, to have been a popular (or maybe not so popular) myth that I must have picked up from a misheard geography teacher or mathematics teacher or else some bloke down the pub way back. On examination, and working from the poles, phi would land at 21.246 degrees (north or south of the equator) while the tropic lines are at 23.5 degrees. That's close but not spot on. And if we tilt the globe at 90 degrees so as to dissect the planet along the equator (so that the polar longitudes and the equator effectively switch places) then, in this scenario the arctic/antarctic circles (normally at 66.5628 degrees) would now be at 23.4372. So again, it's close but still off.

I know I shall regret this  even before I start but here goes. It seems to me that the golden ratio is our human 'thing.' Much therefore as hexagonal combs for bees, coral formations, fibonacci seed arrangements in sunflowers are their 'thing.' I suggest that we are programmed much as everything else in nature to adhering to a pattern. I first assumed this long ago when looking at the angles of twigs and branches of each tree species - as a painter. And then - moving on - perhaps it explains why some abstract art is appreciated and some is not. Just taking a load of squares and rectangles slapped on a back cover and arrange them - some arrangements satisfy others don't. So, I suggest that humans are 'comfortable' with the golden ratio without working out the maths; we are constructed  that way - the perfect form, that is - and it goes someway to explain what we accept as bodies beautiful.
Holds firm and now prepared for board-flak (in that OMG what's P. on about now, mode.)

A bit like this perhaps:



Is the Golden Ratio reflected in Leonardo's Vitruvian Man?  ... I honesty don't know, though I thought it was based more on multiples of 2, 4, 6.

PS

Ah ha! There's an interesting discussion of the maths here:

http://www.crl.nitech.ac.jp/~ida/education/VitruvianMan/

In Leonardo's drawing the ratio of the radius of the circle to the side length of the square is 0.6088, whereas the true golden ratio is 0.6180 .... so the Vitruvian Man, as drawn, is close but not exact. The figure can be redrawn to exactly represent the Golden Ratio and hardly looks any different (it requires the width of the torso to be very slightly widened and the navel raised a tad):



In the above the lines for the square and circle are red for Leonardo's drawing, and blue if corrected to get the Golden Ratio.

Did Leonardo intend his drawing to represent the Golden Ratio, and he just over-simplified the proportions that he states in the accompanying text (and let's face hardly anyone will exactly conform to all these proportions, ratio of height to arm length, ratio of head length to waist height etc.)? Leonardo did not have the benefit of using decimal notation and so had to do everything in fractions or by proportional geometry which could easily give an "error" of the amount we're talking simply by it's imprecision.