Almost the most famous painting (Mona Lisa/La Gioconda/La Joconde) in the world by Leonardo De Vinci...Also uses golden ratio !

Dear Prof Sinan,

Read more about golden ratio in the following links.

http://www.goldennumber.net/golden-ratio/

Certain numerical harmonies, manifested visually in images or aurally in music, seem to resonate harmoniously with the workings of the human nervous system. Perhaps this is because we are part of nature, in spite of modern life sometimes distracting us from being aware of this essential reality. Your question is excellent, Sinan. I don't have the answer, but I am hoping that someone who has looked into the neuroaesthetic dimensions of this topic will provide information that can shed light on this important aspect of our species' appreciation of harmony in art and in nature.

Sinan: I checked my working bibliography for my seminar "Art in the Embodied Mind," and a found a couple of articles that are related to the issue of numbers and aesthetics that I downloaded nearly a year ago; they may be of interest to you:

Moskowitz, Clara (2014). “Equations are art inside a mathematician’s brain,” in Nature (Nature Publishing Group) (http://www.nature.com/news/equations-are-art-inside-a-mathematician-s-brain-1.14825, access: 30 January 2015).

Zeki, Semir; Romaya, John Paul; Benincasa, Dionigi; Atiyah, Michael F. (2014). “The experience of mathematical beauty and its neural correlates,” in Frontiers in Human Neuroscience (Frontiers Media), vol. 8, article 68 (http://journal.frontiersin.org/Journal/10.3389/fnhum.2014.00068/full, updated: 13 February 2014, access: 30 January 2015).

For more on neuroaesthetics, you may consult section 6 of this bibliography, which I uploaded to ResearchGate a few months ago:

https://www.researchgate.net/publication/280949470_Art_in_the_embodied_mind_a_bibliography_%28updated_August_14_2015%29

Data Art in the embodied mind: a bibliography (updated August 14, 2015)

A brief look at the simple symbolic (algebraic) equations of iSpace theory (please kindly see "iSpace - Exact Symbolic Equation for Important Physical Constants - SI Table" on my ResearchGate home) and there especially the derived relations for electron charge e and Planck constant h hinting on how deep nature seems to be interwoven with this "number". I see Golden Ratio more from the point of its geometrical attributes than its numerical value of course, there are many relations to its square and other powers.

Conference Paper iSpace - Exact Symbolic Equations for Important Physical Con...

I would like to suggest a more practical answer - although not to detract from the sense of beauty that arises.

Vision is all about detecting differences. It is also about interpreting those differences in terms of 'objects'. We are told that our experiences of the world can only accommodate a small number of objects - perhaps 4-7. Moreover, it seems that rather than having a fixed capacity, numerating objects is a procedure that 'dies off' progressively as we move on - say from 4 to 7. The last one or two objects require favourable conditions to specify. 

So maybe when our brain is confronted with a visual field input from retina via brainstem the first task is to find the 'main player'. And initially tracking mechanisms may put that centre field. But if the scene is about a relationship we want to identify the second player - the teapot that will pour into the main player cup. And we want to judge the difference in space and retrack to a useful position that still has the main player in commanding position but has the second player in frame too. 

Exactly how this would play out is not clear to me but if the brain handles this sort of information in a statistical way, using multiple comparisons and extracting the most salient answers then it seems to me plausible that the golden ratio is the 'perfect placing' for a brain that wants to numerate objects in a hierarchy of relations, with main player, second player and then maybe third, fourth, fifth player. This might not just be placing in a spatial array along a linear axis. It might be that the amount of resources for object detail is allocated in a decreasing scale, again according to golden ratio. Main player gets 100 cells tuned on it, second player gets 62, third 38, etc. If we need a fourth player maybe main player gets 62 and second player 38 and so on. Some sort of probabilistic competition in a nerve net could make this fall out automatically perhaps. 

Moreover, in reality our attention to objects most often goes no further than main player and second player. But we may the switch to considering the second player with a third player. The golden ratio automatically means that the 'dominance relation' remains constant. And in art it is rare for the artist to play on more than the first two or three lengths in the series, and when someone like Piero Della Francesca does do that we have a sense of shifting into new contexts and relations as we go deeper into the picture.

Another aspect of this triage into objects is that there are reasons for thinking that it might be achieved by shifting temporal phase relations between banks of cells dealing with separate objects. Even there, the golden relation might be a useful statistical rule to use, with increasingly minor players being squeezed into smaller phase ranges. That I am less happy about, but maybe there is some general mathematical relation that governs efficient use of finite resources in all these contexts.

So a picture or building with golden proportions would look harmonious because it would slot neatly into the way the brain likes to divide up the world. Which would be similar to harmony, since the brain divides sounds on the basis of harmonic relations, often inferring a non-existent pitch from associated harmonics.

One of the reasons (and it is not the most important) is that the golden ratio and its "conjugate" appears very often, sometimes in an unexpected way. It is connected not only to algebra, but also to differential equations and geometry.