What thrills mathematicians, inspires artists, electrifies authors and has captivated at least one cosmetic surgeon?
1.6180339887 …
The number is phi, also called the “golden ratio,” the “golden section,” the “golden number” and a host of other terms. People have studied, probed, played and nearly worshiped it for at least 2,000 years.
They have argued about phi too.
Do people enjoy the Mona Lisa, the Greek Parthenon or the smiling face of Cameron Diaz because each revolves around phi? Does phi have anything to do with these exemplars of beauty? Even if it does, could the number somehow take credit for the enthusiastic way people respond to them?
The meaning of phi has engaged people around the world for centuries, and the quest charges ahead into the 21st century. Some declare the ratio an instance of mathematics that explains different aspects of existence, including the appreciation of art. Critics embrace the number as a compelling piece of mathematics, but dismiss notions that it underlies anyone’s feeling of awe during a long gaze at a van Gogh.
Either way, phi is a number unto itself, probably the only ratio with the power to summon and swell its own ideological, spiritual and commercial ecosystems.
It beguiles. It surges with simplicity and complexity at the same time. It’s an enigma.
It inspires. The book “The Da Vinci Code,” which has sold more than 25 million copies, fixed on phi. The movie, to be directed by Ron Howard, is scheduled to hit theaters next year.
The book begins with a murder mystery and swims in international intrigue, none of which describes Denver sculptor Theresa Stroup Ferg’s attachment to phi.
It wasn’t until she was in her mid-40s that Ferg, now 51, first encountered phi. The number charmed her, persuading her to spend the next five years deep in foreign territory – mathematics – in pursuit of a master’s degree in Integrated Sciences at the University of Colorado at Denver.
Her master’s thesis, which she completed this year, centers on a study of people’s responses to “golden ratio” images. The study, she says, suggests the evidence for connections between the golden ratio, which she calls dynamic symmetry, and art are strong.
For the study, she placed series of shapes – like rectangles or stars – in front of volunteers and with each type of shape, one of them would be based on “golden-ratio” proportions.
Study participants were asked to select which rectangle, star and other shapes were the most pleasing. The golden-ratio shapes were the most popular among the participants.
“The information from the study helps us to understand that dynamic symmetry exists in nature, as well as in humans, and this knowledge can guide us to better understand ourselves,” wrote Ferg in an e-mail. “This current study is strongly supportive of the original premise that the human aesthetic appeal is connected to the mathematical concept of dynamic symmetry.”
Ferg and Libby Sales, a CU graduate student colleague studying education, joined together and started The Awakenings Institute, a local nonprofit that aims to introduce aspects of the golden ratio to students in the Denver area. They began teaching an evolving curriculum about three years ago in a Jefferson County elementary school and have since broadened the program to other Denver public schools.
The idea is to fold together mathematics, art and science so that instead of receiving the subjects in different blocks during the day, students get it all at once. This sort of unity is integral to the golden ratio, and its lessons should be broadened into the art of instruction, Sales says.
“Theresa’s study is the first piece in trying to understand why teaching in this manner is just important, for education and the emotional health of students,” says Sales, who hopes either to teach the institute’s curriculum full time in the fall, or get a job as a teacher in public schools.
Ferg teaches sculpture and now uses the golden ratio when she’s showing students how to transform blocks of clay into heads and faces.
“My notion about doing art was Michelangelo with a big block of rock and a chisel,” says Ellen Stevens, an educational psychology professor and administrator at CU-Denver who has taken Ferg’s class. “To work with Theresa to actually see how much the mathematics were there, it was an eye-opener for me. … I think she’s a renaissance woman, frankly.”
People respond most favorably to faces, shapes, buildings, paintings and other things that are based on the ratio, advocates say. Golden ratio studies of Leonardo Da Vinci’s art are the most common – phi proponents hail Da Vinci as a prophet of phi – but many others have been coralled into the phi pen, including the French painter Georges Seurat and the Renaissance sculptor and painter Michelangelo.
The number, phi lovers argue, speaks to something deep inside everyone. Some phi fanatics also claim the mysterious number must be a divine gift.
Others attack notions of phi as a universal aesthetic code as hocus-pocus.
“Almost all of the studies are spurious,” says Stanford University mathematician Keith Devlin, the author of the recent book “The Math Instinct” and National Public Radio’s “Math Guy.” “It is true that there is a strong connection between math and aesthetics and beauty. You can look at symmetries. There are connections. You can look at things we find beautiful and analyze them mathematically. Math has its own beauty. They go together in many ways. Unfortunately, many people whose knowledge of math is very basic … extract from the idea of math and beauty some simplistic notions like the golden ratio, which they wishfully impose onto the world around them.”
University of Maine mathematician George Markowsky even dedicated at least one scholarly paper to debunking golden-ratio claims.
In “Misconceptions About the Golden Ratio,” Markowsky writes that statements about the golden ratio “have achieved the status of common knowledge, and are widely repeated. Even current high school geometry textbooks … make many incorrect statements about the golden ratio.”
“It would take a large book to document all of the misinformation about the golden ratio,” he writes.
Physical echoes of phi are not things we respond to more powerfully than anything else, Devlin says. But the persistence of people’s embrace of phi as aesthetic foundation “says to me that deep in our psyches, we know numbers are very important. They shaped our entire world. Numbers, when they were invented, changed the whole world.”
There are numbers, however, and then there’s phi, the engine behind a blizzard of books, research papers and websites, as well as at least smatterings of paintings, sculptures, buildings – even flower arrangements. What is it about phi that has enchanted people since its discovery?
Take a line. There will be one spot roughly two-thirds of the way down that will divide the line into golden-ratio proportions.
Break the line at the golden ratio spot. Compare the size of the larger line to the whole line – both pieces as they were before being divided. Now compare the larger of the two lines to the smaller line.
The ratios will be the same, an irrational number, one that cannot be captured in a fraction and thus has no end. Digits proceed infinitely after the decimal point, without falling into patterns.
One of phi’s hallmarks is that its application in the physical world lets things grow and contract infinitely while retaining their geometric proportions.
A five-sided star based on phi, for example, will maintain its proportions, regardless of whether it expands to the size of an elephant or shrinks to something an ant could carry on its back.
Phi is the only ratio that permits this infinite growth or diminution while keeping shape static.
Nautilus shells, the arrangement of seeds in sunflowers and pine cones, DNA molecules – all of them and many more examples from nature exhibit manifestations of phi.
While there are other irrational numbers, few if any are as compelling as phi.
“The golden ratio is an irrational number, but it also is in some sense the most irrational of irrational numbers,” says Mario Livio, an astronomer with the Hubble Telescope in Maryland and author of the 2002 book “The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number.”
“This number is extraordinarily fascinating,” he says. “You see pi (in contrast to phi), for example, is a very fascinating number. It appears in math and the sciences, but it has not crossed over into the arts.”
Great debate swirls around the existence – or not – of golden-ratio principles in works of art. Phi promoters say they have found the ratio in a variety of works of art, from ancient buildings to modern paintings.
Critics say claims of phi in many works of art are exaggerated. Phi patrons force their favorite ratio into everything they examine, critics say. In reality, they say, few works of art are based on phi.
Phi was employed in the design of the Villa Emo, a classic Italian villa built in 1559, says Rachel Fletcher, an independent interior-design consultant and scholar in Massachusetts who lectures on the use of golden-ratio principles in design.
“You see it at every level of subdivision,” she says of the villa. “You see it everywhere. It’s quite precise.”
She uses the golden ratio in design because “it gives you a sense of well-being, a sense of, if you enter a room, all of the individual facets have a relation to one another,” she says. “When you are in an environment where that is taking place, you feel the world is right, nothing is out of place.”
Fletcher does maintain that phi probably stirs everyone in mysterious ways. She has followed debates about phi for 25 years, she says, and doesn’t spend much time anchored to either camp.
“The fact that it has persisted for centuries over history, I think, is very compelling,” she says. “If it isn’t a wonderful proportion, it’s one we all want to be a wonderful proportion.”
The proportion is wonderful enough to Connecticut architect Frank Ryan that he named his firm not Ryan Architecture Inc. or Frank Ryan LLC, but The Golden Mean Group.
The gesture, though, doesn’t mean Ryan believes the pleasures of phi have anything to do with biology.
“For me it’s a constant reminder that our lives are composed of balance and proportion,” he says. “When I refer mentally to the golden proportion, it’s sustenance to understanding our relationship to the natural world.”
Ryan does not believe in the universal appeal of phi, but he does think it probably resonates at least in some people.
“I think what is hard-wired with each person is their individual sense of balance and proportion, which might be represented by the golden mean,” he says. “But there may be square people and thin rectangle people.”
And nose people and chin people. Stephen Marquardt, a retired cosmetic surgeon in Huntington Beach, Calif., has dedicated years to the study of the golden ratio in the human face, and now in the body. He has appeared on The Learning Channel, the Discovery Channel and other media outlets to talk about his research.
Frustrated with an absence of scientific answers to the question “What is beauty?” Marquardt set out to find keys to physical attraction. His search quickly led to the golden ratio, and he immersed himself in phi.
Using a complicated set of golden-ratio measurements and proportions, Marquardt created a beauty “mask” that shows where, for example, eyes will ideally sit in relation to the tip of the nose. The mask, he says, is an objective and scientific portrayal of universal beauty.
With the mask as a benchmark, he says he can study a woman’s face and quickly understand what does not fit within the golden ratio. This makes decisions about what to do with lips or noses much more easy.
He even received a patent, in 1999, for the mask.
“When I went to the copyright office it was fun,” he says. “Nobody had tried to copyright beauty before.”
He’s no phi-as-intrinsic evangelist, but University of California at San Diego psychology professor Vladimir Konecni still might respond with pleasure to a face transformed by Marquardt’s vision of golden-ratio perfection.
Konecni dedicated years to studying intersections of the golden ratio and aesthetics, and he celebrates the number, which he says is one proportion among others that translates into pleasing aesthetics.
It’s not built into our genetic code. Instead, the golden ratio is “something you have to think of in art as an interesting proportion that was passed from father to son, teacher to pupil, over the centuries,” he says.
His own art photography, in fact, revels in the golden ratio.
“If you get an invitation to one of my art shows,” he says, “you get a golden rectangle.”
Ferg intends to continue studying the golden ratio. She’s on a waiting list for a Ph.D. program at CU, she says, and other scholars are helping her get her master’s thesis published.
Much more research is needed, she says, before important questions about the golden ratio and aesthetics can be answered.
“I came to it as an artist, and I still am an artist,” she says. “What I think has held this up a little bit is that artists don’t necessarily document well what they do. So I’m trying to change that in my own procedure here. I’m going to be painting some portraits here coming up, and I will carefully measure these proportions out on graph paper before I start the painting.”
“Rather than just making the art,” she says, “I’m going to take a little time to see if I can get some data.”
Staff writer Douglas Brown can be reached at 303-820-1395, or djbrown@denverpost.com.
