Popular misconceptions about the Golden Ratio (2004)
maa.orgFor the artists and people who do use the GR, myself included, I use it out of convenience when any arbitrary decision will do. It's just an easy way to count exponentially.
For instance, I recently wanted to compute a moving average of recent events. I chose to use Fibonacci (i.e., last 3 days, last 5 days, last 8 etc), but my choice had nothing to do with any belief on my part that it has any deep significance. I just wanted a moving average and wanted decreased granularity as events got further out. I use powers of 2 as well. For instance on server retries after a failed request.
I suspect if the GR is used a lot in art it's done for motivations similar to mine: when we're presented with deciding on something arbitrary we can easily limit our choices and simplify decisions by using an established counting method.
Do you think Shakespeare's poems (iambic pentameter) would be any less beautiful if he decided to go for 12 syllables instead? Would we not like music as much if it were written in 6/4 instead of 4/4? There are limitless choices so we fall back on established conventions simply because any choice will do.
Even the nautilus shell, which this article says follows the golden ratio, doesn't. Proof: m_for_monkey's link: http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
I believe the author got it backwards. I don't think most artists follow the Golden Ratio (at least not consciously, or since this appeared in literature), as much as this ratio has a tendency to creep up over and over on people's works.
I have a theory about this, and it has relation to ocular dominance [1]. People who photograph or shoot firearms might know about this: our brains tend to favor one of our eyes (just like left/right hand dominance), so we have a kind of overlap where the image we see is not 50/50 between each eye. That might point to a mental bias to find slightly unbalanced compositions (that approach the GR) more pleasant than completely balanced ones.
The last sentence - that we find unbalanced compositions actually more balanced - is empirical, not just speculation from my part. Any artist knows that intuitively too.
I don't know if there are papers about that, if you know please comment!
Another good writing on this topic is the Fibonacci Flim-Flam:
A rough idea of why the Fibonaccis and the golden ratio are related: in making a golden rectangle, you take an existing rectangle, attach a compass across its long end, and use that compass to construct a square. You add the square to the golden rectangle to get... another golden rectangle. If you wanted to repeat this, there is also a "flipping" of the rectangle by 90 degrees. Mathematically, the process takes a rectangle (a, b) (where b > a) and produces a rectangle (b, b + a). But that's just the Fibonacci recursion relation, so starting from (0, 1) or (1, 1) you'll generate all the Fibonacci rectangles as you 'goldenize' them further and further.
An actual proof: Since φ and 1 − φ are the solutions to the quadratic x² = x + 1, they also solve xⁿ = xⁿ⁻¹ + xⁿ⁻² and thereby form a basis for all solutions of the Fibonacci recurrence relation: in other words, you can solve the system of equations {A + B = F₀ ; A φ + B (1 − φ) = F₁} for parameters A and B, and then the recurrence relation guarantees A φⁿ + B (1 − φ)ⁿ = Fₙ for all remaining Fibonaccis. (For F₀ = 0, F₁ = 1 this gives A = -B = 1/sqrt(5).) Since |1 − φ| < 1, this decays geometrically to 0 and the dominant term is simply Fₙ ≈ A φⁿ.
The article thankfully does explain that the golden ratio is the most irrational number: because a continued fraction expansion gives the best rational approximations, large numbers in the continued fraction expansion make for extra-good rational approximations, and the golden ratio has the smallest numbers possible -- they're all 1. What it doesn't quite explain is that if you're growing a spiral by spitting out dots and you turn by an angle 2π · α each time you spit out a new dot, those spirals often seem to "line up" in lines given by the denominators of the best rational approximations. So if you have a rational number like α = 22/7, you would literally see just 7 straight spokes coming out of the center; if you use α = π you will see a region where there seem to be seven spirals due to the fact that 22/7 is a disproportionately good approximation to π, off not by ~ 1/14 but rather by ~1/790. Anyway this is the more concrete reason why, if you take a sunflower and count the spirals, you always seem to "magically" get Fibonacci numbers; they're denominators in the best rational approximations.
You may wish to experiment with a spreadsheet to see all of this dynamically for yourself, a-la http://tmp.drostie.org/sunflower.png .
Of interest to hackers! This hacker at least.
More articles like this please. :-)
I like one idea I heard, which is that the pentagram is a symbol of freemasonry because originally it was used by actual stone masons to provide a golden mean for cathedral construction.
But without extremely good evidence, this is nothing but a nice thought, though it does give a fairly practical reason for it gaining the status of an occult symbol, which it didn't seem to have so much before the period of massive cathedral construction.