How significant is the golden ratio? Did DaVinci really use it in his artwork? Is it seen regularly in Nature? Where can it be seen other than sea shells? Yes, I have seen the movie PI…just wondering how much truth there is in the movie.
I’ve never seen a movie as technically bad as Pi, so you may not want to rely on it for too much. That said, the discussion of the golden ratio was as close as they came to getting anything right. You can find a bit more detail here.
And a huge amount more info in The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number, by Mario Livio
Short answer. Most artwork does not in fact use phi. Later commentators imposed phi on the works by fudging. Ratios close to phi are common, and it appears to be a ratio that does work well aesthetically, but few use it deliberately and systematically.
Here’s my “golden ratio” story: I was first introduced to the golden ratio in elementary school by the Disney educational movie Donald in Mathmagic Land. Fifteen years later I had the opportunity to use it. Right out of college, I was employed by a defense contractor. They had just purchased land to build a new manufacturing facility and 1000-meter firing range. One day my boss came to me and said, “Mr. [Owner of the company] is driving us crazy. We’ve submitted a zillion proposed layouts for this new facility and he hates them all! He can’t say what’s wrong with them except that “they don’t look right”. I can’t deal with this any more so I’m dumping the project on you. Have fun!” The first thing I did was to draw the entire property as a golden rectangle. Then I drew a line across the shorter span such that the rectangle was divided into a large square and a smaller rectangle, which was also a golden rectandle (they showed this in the Disney movie). I put the 1000-meter firing range inside the large square, and I put the manufacturing, office and ammo storage buildings into the small rectangle. I also divided the small rectangle again into a square and even smaller golden rectangle. I put the manufacturing and office buildings into the square and the ammo storage buildings into the smaller rectangle. The lines of division were also physical things like roads or fences. Anyway, I showed this layout to the Owner and he approved it on the spot. That’s how I first got myself noticed at that company. Three years later I was Chief Engineer.
And did you label it φ-ring range? 
Another good book is H.E. Huntley’s The Divine Proportion:
I learned about phi from an old Science Digest article in my uncle’s basement, and from the Disney short Donald in Mathemagic Land, both of which point out the many placves where phi shows up. It’s impressive and widespread, but I have become disillusioned with the phi-worship I see. think that too many people see i in places it really isn’t, and it’s not quite ubiquitous. The 19th century research that supposedly showed that a rectangle with ratio of phi:1 is the most aesthetically pleasing has been roundly criticizede recently. I’ve observed for years that, of all the places we use rectangles – TV screens, cinema screens, paperback books, hardcover books, postcards, writing paper (American, European, Legal), money, computer cards, business cards, —not one example has the Golden Ratio. If it really were some sort of “most Aesthetic”, you’d thibnk that someone would have exploited this feature by now. I’m similarly suspicious of its oft-cited appearance in art. If you choose enough examples, you’ll certainly find cases of golden trectangles. But you’ll find lots more that aren’t. You’ll find lots of spirals, but they won’t all be logarithmic. In particular, those spirals in the heads of sunflowers and the like hasve recently been argued not to be logarithmic at all (See The Penguin Dictioary of Curious and Interesting Geometry, and references thyerein.)
I think what happened here is that someone noticed that sunflowers have a connection to the Fibonacci Sequence, and that the Fibbonacci Sequence has a connection to phi and to spirals, and that sunflower heads have spirals in them, and just jumped to the (naïve) conclusion that those spirals must be the same. They aren’t.
phi does show up in a lot of places, though. It can be very useful in pseudo-random sampling, for instance. If you take a circle (a pie, say), and keep making radial cuts, each one an angle of phi*2pi from the last, then at any given moment, you’ll have cut the pie into approximately-equal-size slices (each slice will divide the currently-largest piece into two pieces smaller than the current smallest).
By the way, at least two articles arguing (as I have) that phi is not as ubiquitous as it’s often held to be have appeared (although I reached my idea independently). One was in **The Skeptical Inquirer within the past five years. The other was an article by Willy Ley (who’s been dead a long time now, so it’s an old article) arguing against people who saw weird number correlations (including phi) in the proportions of the Great Pyramid of Khufu/Cheops. It appeared in a very obscure anthology of his writings, but I can’t recall the name right now.
I’d like to second the recommendation for Mario Livio’s book. Great stuff. Pretty early on, he mentions a great number of places one can find it, but also a great many places where you can’t. I remember being a little disappointed that the pyramids, the Parthenon, and my boob tube were not so perfect after all.
But people do exploit the ratio for aesthetic purposes. When I was in (music) college, one particular teacher had a real hard-on for Fibonacci. He would make students write compositions featuring the ratio. My roommate wrote a jazz comp where the climax occurred exactly .618 of the way through. Impressive though that sounds, how can you hear the wonderfulness of that when you are unaware of when the song ends? “Wow, that sucked, but man, just a little more than halfway through, it was awesome!”
The Library of America’s books add to their super-coolness by being trimmed (as nearly as reasonably possible) to the Golden Ratio: 4.875 x 7.875 inches.
I’ve done a quick measurement of other books, both hardcover and paperback, and none are quite to that ratio.
Interesting, G.U.. As I observed in post #7, standard paperbacks, trade paperbacks, and most hardcovers certainly don’t fit the golden ratio. This is the first example I’ve heard of that does.
You’re φred.
Erno Lendvai, a musicologist, wrote a book about how Bartok shoved the golden section into lots of his compositions, and he seems to have a point. Bartok even seems to have used it recursively. So a movement would have some climactic bit 0.618 of the way through (in beats rather than seconds IIRC), but the first section would be divided at its 0.618 point too, and so on.
Of course I pondered that for about 0.618 seconds before having the same thought you did. I suppose one might apply one’s appreciation retrospectively. At the climax one might subconsciously determine “aha, the piece must carry on for just this long to be proportioned wonderfully”, and then be pleased when that in fact happens… but it sounds like codswallop to me.
Bartok’s one of my favourite composers, but certainly not for what seems to be his mystical misapplication of a piece of mathematical candy.
FWIW, credit cards and bank cards are close to the golden ratio.
Some common sizes of drafting paper are close to phi: 11 x 17 (ANSI B), with a typical 1/2 inch border, gives a drawing area of 10 x 16. Also size 22 x 34 (ANSI D), but not as close. Note that half of an 11 x 17 sheet is 8.5 x 11 (ANSI A), while two 11 x 17 sheets makes 17 x 22 (ANSI C).
I just measured mine, and got 2 1/8" by 3 3/8", for a ratio of 1.588:1 . The golden ratio, meanwhile, is 1.618:1 , or about 1.8% off. Close, maybe, but if they had wanted to, they could have gotten considerably closer.
iPods were designed with the golden ratio.
Another curious fact about φ is that it’s the number whose square is one more than itself. How that relates to all its other attributes I have little, if any, idea.
Yep. φ^2 = φ + 1. As well, 1/φ = φ - 1.