What proportion is still golden mean (phi) ratio and what not yet?

The golden ratio, by definition, is the larger of the two roots of the equation x2=x+1x^2 = x + 1, namely \phi = \frac{1 + \sqrt{5}}{2}.  Any claim that it represents any kind of physical ideal is, in my opinion, absolute nonsense.  There's no fundamental reason to believe that it would be particularly suited for any such thing.  Most people have heard that the ancients believed this number to be particularly aesthetic and that, say, the Parthenon was built to this proportion, but these stories are simply false.  The truth is that people like things that aren't exactly square, but aren't so much as twice as long as they are wide.  \phi is just a number roughly halfway between 1 and 2, and if you used 1.5 instead I doubt it would affect anything.

See Daniel McLaury's answer for the definition of the golden ratio. Also, as he mentions, there is a lot of numerological nonsense surrounding the golden ratio. The reason that it is mathematically significant is that it shows up in geometry and number theory. It is the chord of a regular pentagon with unit sides, and is the circumradius of a regular decagon with unit sides. It also related to the Fibonacci and Lucas numbers. The ratio of consecutive Fibonacci numbers, F(n+1)/F(n), approaches the golden ratio as n approaches infinity. It shows up in nature to the extent that the Fibonacci numbers show up in nature, but any suggestion that it is tied to some deep law of nature is nonsense. People tend to mysticize ratios in nature which come close to the golden ratio. For example, the proportions of various parts of the human anatomy, such as the segments of the finger, are often asserted by cranks to be in exact proportion to the golden ratio, but very few people, if any have these exact proportions, and contrary to the assertion that such proportions are more attractive, they would probably actually seem a bit unnatural (kind of like when you look at the proportions of a life size Barbie doll).

"Hidden harmonies are more than obvious." Heraclitus I've recently shared on TED how the golden ratio can also be seen as a temporal signature (an asymptotic convergence, to be precise), of dynamical systems universality, of how nature evolves emergent complexity most robustly, adaptively and rapidly. But really, looking exclusively at only Spatial interpretations are so last century anyway. We need to look at what nature is doing in Time. And does that also mean the entire repertoire of plant phyllotaxis is an illusion? Plants are telling us something, that we need to think harder about the Temporal, thermodynamics and morphogenesis, as Alan Turning did in his last research on the Fibonacci series, phyllotaxis and morphogenesis before he tragically died at an early age. We need to look at both the animate and inanimate, at the dynamical and space-time, not just the frozen and static. It seems the way nature works is to manifest the golden ratio not as a spatial fossil, rather an optimal temporal energy flow signature. An efficiency (& beauty) constant. All this current argument is only over the residue or sedimentary forms of these dynamical flows - so we are missing half the picture (as I've been saying since first publication in AD Magazine, based on Masters studies in Architecture & Computer Science at the University of Westminster, London). https://asynsis.wordpress.com/2014/09/05/entropy-begets-design-qed-2/ What the sciences of complexity and dynamical systems have been sharing recently is that the golden ratio is one of several optimal, analogical geometric signatures of how nature evolves emergent complexity most easily - over time. It's a dynamical behaviour, a verb not a noun. It's not the phone number, but the very action of dialing. Here's how I Tweet it: #Asynsis #DaoOfDesign on #TED at #TEDxWanchai #HongKong. A New, Extremely Lean, Mean (#Design) #TheoryOfEverything http://tedxtalks.ted.com/video/Form-follows-flow-%7C-Nigel-Readi;search%3ANigel%20Reading http://asynsis.styleonedigital.com/archives/4017 http://about.me/asynsis The very latest examples (including E8 & ER=EPR #Universality), are shared here (enjoy!): https://www.facebook.com/pages/AsynSo%CF%86ia/202383966558282

Like others, I don't see it as any kind of physical or esthetic ideal. It is found in the Fibonacci series as the limit of the ratio of adjacent number as go up the series. The Fibonacci numbers appear in nature in the spiral growth of seeds or offshoots from a stem. Examples are the placement of palm fronds on the tree, the flora on broccoli and cauliflower, the ratio of the number of rows in the pineapple as they spiral in different directions (typically 5:8, as I remember), the rows on a pine cone. Why does nature spiral growth from a stem at 0.618 of a circle (or 1.618, if you wish)? The most obvious reason is that phi is an irrational number so no new blade of grass or palm frond will be directly below another, thus getting less sunlight. Placing the next growth at 0.7 of a circle would result in the 11th shoot being directly above the first. Beyond being an irrational number, I'm not an expert but my understanding is that it facilitates the exponential growth rates and allows the different growths to fit together without leaving gaps or squeezing each other out.

Though mathematicians around were strict and definitive:), it’s still pretty clear, what you actually mean. So yes, since every occurrence of \phi in the real world is an approximation, which approximation is “good enough” and which is no longer?That depends on the question “good for … what?”Hence, first and foremost must answer, what is the “usage” of \phi?One of the most obvious features and thus “applications” is its ability to generate well-dispersed and disordered, yet “harmonic” (in general sense) distribution. Such as sunflower seeds. Or let’s look at the https://en.wikipedia.org/wiki/Orbital_resonance problem:If two planets rotate around the Sun (only 2 for simplicity), having cycle periods a and b, while affecting each other via gravitational attraction. What would happen?If the ratio \frac{a}{b} equals ratio of two small integers (e.g. \frac{3}{5}, then the point of minimal distance between them would occur at fairly same places (on rotation plane) and fairly regularly, e.g. in case of \frac{3}{5} it’s 2 times for each 5 rounds of the slower planet (or 3 rounds of the faster planet, which is the same).Such regular rhythmic tugging, even small, would work as a swing, increasing declination from initial orbital trajectories, possibly leading to complete disruption of the orbits.Yet if the minimal integers of the ratio are big, thus difference between them is also big, then: the planets in most of periods will have closest passing in the different places on the orbital plane, thus counter-balancing mutual gravitational disruption, adding some stability instead encounters in the same phase (place on the plane) will occur rarely enough, to have no resonant effect Let’s look at Earth and Venus. Their periods are 365.25 and 224.7 days respectively.The ratio is about \frac{224.7}{365.25} \approx 0.6152 or \frac{365.25}{224.7} \approx 1.6255.Pretty close to \phi \approx 1.6180339, yet pretty far also. If the ratio is close enough to \phi, then there is anti-resonance of orbital periods, and no disturbing resonance effect.What are the nearest ratios of small integers? These are: 8/5 = 1.6 13/8 = 1.625 and let’s include also 3/2 = 1.5 The nearest is of course 13/8, and basically it is the ratio to decide - is there resonance or counter-balance?For certain, 3/2 = 1.5 is a resonance and thus orbits won’t be stable (that also depends on the actual forces magnitude, but generally it’s like this).The ratio of 8/5 = 1.6 would be more stable but also orbital resonance might occur (again, depends on actual forces).13/8 = 1.625 is already a stable configuration in this case. The planets meet 13 - 8 = 5 times, every 8 Earth years, in “almost pentagon” corners, which gives sufficient counter-balance of mutual attraction. It’s not unlikely that Earth and Venus positioned themselves this way around \phi, exactly because of the “oasis of stability” in terms of ratio approximations. (see “golden ratio is most irrational number”)The perfect golden ratio would provide perfect disordered distribution and thus “perfect counter-balancing”. In practice, the oasis of stability usually occurs between 13/8 = 1.625 and 21/13 \approx 1.61538.So the interval where “it’s still fairly \phi” is pretty wide.PS. For celestial bodies, the example above is a simplification, since with more than 2 bodies, more complex relationships play role (e.g. Laplace resonance 1:2:4, which is stable despite small integer ratios).PS2. Depends on the usage! In other cases other criteria for \phi-ness might play role.