Oh dear, here we go again: the Divine Proportion, the da Vinci code, the Pythagorean Brotherhood, Golden Rectangles. They all come down to single number, the Golden Ratio φ. Why does it mesmerize us? Not because it's irrational; so is √2. True, it has some curious properties, like φ^ = φ + 1, and you find it in Fibonacci numbers and Penrose tilings. But I think it mesmerizes because it's evanescent, mysterious: now you see it, now you don't. The golden ratio shimmers in spirals of pine cones and leaves on a stem and in the façades of ancient architecture. But when you look closely, it isn't there. The golden ratio is Exhibit A for mystical thinking.
I muttered thusly on reading a recent article on "why we love beautiful things." Why, the author asks rhetorically, do books, television sets, credit cards and the original iPod have the same approximate shape? Because, he goes on to tell us, these rectangles are (almost) golden. Get out your ruler and measure a few. You'll see: their length to height ratios are around 8:5. That's 1.6. The golden ratio φ is 1.618 to the first three decimal places.
Lance Hosey, the author of the article that bestirred me, isn't talking about pine cones or the Parthenon, he's talking about modern design. Books, television sets, credit cards and the original iPod have the shape they do because people like it. "Experiments going back to the 19th century repeatedly show that people invariably prefer images in these proportions," he tells us, "but no one has known why." No one, that is, until a few years ago when "a Duke University professor demonstrated that our eyes can scan an image fastest when its shape is a golden rectangle."
Really? Is this fact? Or is it imagination? Skeptical, I googled and clicked and found that paper. I learned that it's something of both.
The Duke professor is Adrian Bejan. His paper has the title "The Golden Ratio Predicted: Vision, Cognition, and Locomotion as a Single Design in Nature." So there it is: golden. But he doesn't predict the golden ratio! Instead, he explains why the golden ratio is not the holy grail. Searching for physical principles behind the number φ is misguided, he says. Approximate golden rectangles are the objects of interest: "The physics phenomenon is not φ itself . . . The physics phenomenon is the emergence of shapes that resemble φ." Professor Bejan's experiments show that "humans scan the world on a two-dimensional screen approximated by a rectangle with the shape L/H ~ 3/2." That's a fact, and savvy designers know it.
But it takes imagination to see this shape as golden. It's not even close. Divide three by two and you get 1.5. Subtract 1.5 from the golden ratio, 1.618, and you get 0.118. That's a big difference. The orbits of the planets are more circular than that: Venus's eccentricity, on a scale of 0 to 1, rounds off to .00678, Earth's to 0.0167. Indeed, φ - 3/2 is greater than the eccentricities of all the planets but Mercury!
Eccentricity matters. When your GPS says turn, you glance at the screen and do as you're told. The screen is shaped so a quick glance does it; you wouldn't want it to be square. The disembodied voice brings to mind the old movie, "God is my co-Pilot," but a GPS isn't a deity, it's a receiver synthesizing information from a flock of satellites circling the earth. Correction: orbiting the earth, not circling it. That screen had better be accurate! As another World War II phrase had it, "loose lips sink ships." Updated, that's "loose math, quick crash." Recalculating . . . recalculating: the GPS receiver needs to "know" exactly where each satellite is at every instant. If it "thinks" they're sailing in circles, you'll end up in a ditch.
In his youth, the great seventeenth century astronomer Johannes Kepler agreed with his contemporaries: planets move in perfect circles. Well, not quite perfect, but almost, and circles are beautiful and beauty is truth. Then he discovered that "almost" was ever-so-slightly elliptical. We tend to forget that the Copernican revolution was a two-stage affair; first heliocentricity; then eccentricity. Kepler freed astronomy from its self-forged circular chains and changed the human mindset.
If orbits stopped being circles in 1609, why are we still talking about golden rectangles in 2013? I suggest it's a problem of naming. Kepler had a name at hand; he'd learned about ellipses in Euclid's Elements, Book XI. To break the gold chains that still shackle rectangles, we need a new name for the rectangles we care about, the ones like our window on the world. Let's call them brass rectangles. A rectangle will be brass if its L/H ratio lies between 1.5 and 1.736. (I got that upper bound by adding 0.118 to φ. It makes φ the midpoint of the interval and includes my GPS screen, whose L/H ratio is 1.722.)
"Brass rectangle" has qualities a name ought to have, if it's to catch on world-wide. "Brass rectangle" is evocative: brass looks golden if you polish it. It has historical gravitas: brass was smelted in ancient Rome. Most significantly, brass isn't an element, it's an alloy, a mixture of copper and zinc. The copper:zinc ratio varies from brass to brass, with many if not most in our rectangle range!
Why do we like beautiful things? If the Brain Activity Map project gets off the ground, we'll be reading more about this thought-provoking question. I always took the expression "beauty is in the eye of the beholder" to mean that beauty is subjective. But maybe "like" and "beautiful" are synonyms under the skin. On the research frontier where biology, psychology, and mathematics meet, the eye of the beholder and the beholder's brain and genome are an integrated system, a system to be mapped. I hope the golden rectangle won't be a map symbol.