## What are some arguments for the golden ratio making things more aesthetically pleasing?

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What are some (not necessarily good) arguments that painters, architects, designers, musicians, etc. basing their work on the golden ratio φ makes their work more aesthetically pleasing?

I think these lines of reasoning are heard the most:

(a) The golden ratio can easily be found in nature [I include mathematics as part of nature, thus including all instances when the constant crops up, but this can be relaxed] in a lot of unrelated and unexpected places. Therefore, the constant is part of nature.

Anything made relating the thing to nature in some way (e.g. using a material found in nature: a wooden chair, a golden ring, etc.) is always more aesthetically pleasing because nature itself is the epitome of aesthetic beauty, therefore always adding aesthetic value to the thing made.

(b) The simplest visual representation of φ, the golden rectangle, is aesthetically pleasing; same goes for other simple geometric figures based on φ.

(c) All classic art is aesthetically pleasing. There is classic art incorporating φ, therefore there is aesthetically pleasing art involving the golden ratio.

Note: I am not talking about art exclusively.

The golden ratio has an important political interpretation as well; consider one way to state the geometric definition: the greater is to the lesser as the whole is to the greater... – Joseph Weissman – 2016-10-05T20:42:36.397

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I'm not sure of the philosophy behind it, but so of Da Vinci's work related proportions in the (ideal?) human form to this ratio. https://en.wikipedia.org/wiki/De_divina_proportione

– Dave – 2016-10-05T20:56:51.657

DaVinci attributes these to https://en.wikipedia.org/wiki/Vitruvius#Proportions_of_Man

– None – 2016-10-05T21:22:45.800

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More often than not artists do not give arguments about using the golden ratio, they are sufficiently motivated by the long tradition of singling it out as "golden", which accumulated since Pythagoreans and the ancient Greek sculptor/architect Phidias. The perceived presence of golden ratios in his Parthenon now appears to be spurious, but the letter φ often used to denote it comes from the first letter of his name. Plato helped too by promoting dodecahedron, "which the god used for embroidering the constellations on the whole heaven", and which is full of golden ratios just like the Pythagorean pentagram. Both Pythagoreanism and Platonism flourished during the Renaissance, when much of our modern artistic tradition was forged, and even Copernicus and Kepler were self-proclaimed Pythagoreans. This said, the extent of the ratio's use in arts is greatly exaggerated today. Here is Falbo:

"It is remarkable that prior to Fischler’s and Markowsky’s papers, there seemed to have been no set standards for obtaining measurements of artwork. Often, a proponent of the golden ratio will choose to frame some part of a work of art in an arbitrary way to create the appearance that the artist made use of an approximation of φ. Markowsky shows an example in which Bergamini arbitrarily circumscribes a golden rectangle about the figure of St. Jerome in a painting by Leonardo Da Vinci, cutting off the poor fellow’s arm in order to make the picture fit."

Livio in The Golden Ratio and Aesthetics is similarly skeptical:

Many books claim that if you draw a rectangle around the face of Leonardo da Vinci's Mona Lisa, the ratio of the height to width of that rectangle is equal to the Golden Ratio. No documentation exists to indicate that Leonardo consciously used the Golden Ratio in the Mona Lisa's composition, nor to where precisely the rectangle should be drawn. Nevertheless, one has to acknowledge the fact that Leonardo was a close personal friend of Luca Pacioli, who published a three-volume treatise on the Golden Ratio in 1509 (entitled Divina Proportione).

The occurence of the ratio in nature is also exaggerated, as Falbo points out, "we find that spirals in sea shells do not generally fit the shape of the golden ratio. This is true despite the numerous articles on the Internet and elsewhere, in which pictures apparently have been stretched to fit". It is also curious that the ratio was not extracted "from nature" before Pythagoreans, whose interest in it was apparently sparked not by nature but by the "mystical perfection" of the pentagram.

This is not to say that the ratio does not genuinely occur in nature (along with many others), or that some artists did not consciously use it, e.g. Dali in the Sacrament of the Last Supper. Le Corbusier, the architect, developed a whole system of proportions, called Modulor, which "was supposed to provide a standardized system that would automatically confer harmonious proportions to everything, from door handles to high-rise buildings."

Aside from the half-fabricated tradition, a boost to this usage was given by Fechner's psychological experiments in 1860-s, where the participants were shown ten rectangles, and 76% chose as "the most pleasing" the ones with the side ratios of 1.75, 1.62, and 1.50, the 1.62 being closest to the Golden one. Trouble is, later experiments did not reproduce Fechner's findings, and Godkewitsch's meta-study in 1974 concluded that the preference was an artifact. According to Godkewitsch:"The basic question whether there is or is not, in the Western world, a reliable verbally expressed aesthetic preference for a particular ratio between length and width of rectangular shapes can probably be answered negatively". More recent experiments concerning proportions of faces also failed to confirm the golden ratio's hype. As Livio points out:

"The history of art has nevertheless shown that artists who have produced works of truly lasting value are precisely those who have departed from any formal canon for aesthetics. In spite of the Golden Ratio's truly amazing mathematical properties... I believe that we should abandon its application as some sort of universal standard for "beauty", either in the human face or in the arts."

Nick Seewald runs a website on golden ratio, which has a lot of up to date information on its occurences, real and mythical.

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There are sources to discover the Golden ratio in arts: for instance History of the Golden Ratio or Golden Ratio in Art and Architecture. From what I have learned, possible aesthetical reasons are largely due to Luca Pacioli who, in the De divina proportione, provides semi-theological motivations, among which:

• unique, as God
• similar to the holy trinity, one substance in three, it is a unique proportion defined by three terms a/b = b/(a-b)
• related to the construction of one of the five regular solid (dodecahedron)

Apparently, people have been hunting φ everywhere, while falling in a selection bias: it is easy to find φ somewhere in a construction, a painting, and disregarding the other proportions. It is easy to find a number close to φ, and pretend the Golden mean is present. Moreover, recent studies or polls have shown than when people are exposed to several triangle or rectangle shapes, the most visually pleasing is not the one with the Golden mean, for instance in Le nombre d’or : réalité ou interprétations douteuses ?.

However, it has interesting merits, on the mathematical side (e.g. Is there any integral for the Golden Ratio?, Is there a “positive” definition for irrational numbers?). For instance, φ is "the" irrational that is the least approachable by fractions. It can be written as:

I doubt those properties have influenced artists.

1very good answer! i like the idea that it's aesthetically pleasing because being the "irrational that is the least approachable by fractions" it answers the most unanswerable questions hah ;) – None – 2017-06-20T22:29:01.870

I personally prefer the definition of phi being the only non-0 solution to x^(n)+x^(n+1)=x^(n+2). It's a little cleaner, and more easily translates to an actual ratio. – Weckar E. – 2016-10-06T05:09:26.273

Is the $x^n$ really necessary, or a trivial "higher-degree" extension? I believe the equation has 2 non-zero solutions – Laurent Duval – 2016-10-06T05:28:26.240

I personally find it a more useful definition (and yes as you point out there is also the minor ratio, but that's not entirely relevant to the discussion - but it's relationship to phi is actually rather cool). The important thing is that this definition implies a growth paradigm, which is the reason it can be found in nature at all. – Weckar E. – 2016-10-06T05:31:57.703

The equation $x^2-x-1=0$ directly follows from the second point from the excerpts from Pacioli. Historically, I am not sure the symbolic notation with unknown $x$ was actually used at that time. The use of letters in equations (in the Western world) seems to be posterior, with the work of François Viète, born 23 years after Pacioli's death – Laurent Duval – 2016-10-06T06:23:37.850

1And now you are going way over my head. – Weckar E. – 2016-10-06T06:37:18.143

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One property of segments who conform to the golden ratio is that they are foldable.

Just look at the way you can "roll up" your index finger snugly inside your thumb. That would not work if the segments of the index finger were all in the same length or if the outmost segment was shorter and did not conform to the ratio.

By being able to do this "snugly" it adds to stability. Stability = good. I'd say.

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If DaVinci's observations about the 'Vitruvian Man' are just empirical trends, this number may represent a natural human allometry. Some aspect of the underlying pattern upon which the distribution of body measures falls on average may just have this as its dimension.

If we like this as a balanced limb ratio, or some other common comparison, when it arises in other humans, we might generalize our response to other visual input.

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As stated above

φ is "the" irrational that is the least approachable by fractions.

It is the number which is least approachable by dividing integers. You could quite easily suppose that this is why, that it resists the

basic operations of arithmetic

and so is in some way the most sublime of ratios. But that would only work if we all already intuitively know that it is, given that the discovery of that quality has not influenced our aesthetic history.

Given there are no perfect golden shapes in out drawings, it seems unlikely that its geometric qualities can help us down this route. How could we all already know the value, anymore than we do pi?

In conclusion, any answer to your question would, of course, depend upon artists etc. just knowing φ, that number which most resists counting numbers in basic arithmetic. While that quality could, I think quite obviously, have psychological implications, I'm not sure I believe that there's innate knowledge of mathematical constants etc.. Perhaps e.g. Leibniz would think we innately know there is φ, but how could we know it is (roughly) 1.618?

It's only really conceivable I think, if we agree with Leibniz's innatism, and add that we all instinctively know the limits of our knowledge. Supposing we do, that could end up generating a lot of differences in philosophy.