Spotlight on Ohlone:
Math connects to art, nature
By Eric Dorman, Editor-in-chief.
Thursday, March 16, 2009—Reprinted from Monitor.
Life, imitates art, imitates...math?
If you thought connection between math and art—or even math and real life—seemed sketchy before you walked into Math Instructor Jeff O’Connell’s Brown Bag seminar on the Golden Ratio last Friday, you probably weren’t alone. But after O’Connell’s hour-long presentation connecting square roots and the quadratic formula to the Mona Lisa and sunflowers, the link between math and art was a hard one to ignore.
Whether you look in your favorite math tome or simply look at yourself in the mirror, the Golden Ratio appears, O’Connell explained before a standing-room-only crowd in 3201. Though derived through math, the Ratio lies at the heart of art, architecture, graphic design, nature and even the human body. “The Golden Ratio is everywhere,” said O’Connell.
O’Connell began the presentation with math. The Golden Ratio, he explained, can be arrived at in a number of ways. The ratio itself is simple: it describes the ratio between two lengths whose sum, divided by the length of the longer one, is equal to the larger sum divided by the smaller one. O’Connell first evaluated the equation using the definition; calling the shorter length 1 and the longer one x and then evaluating the fractions using the quadratic formula. He also demonstrated how the ratio could be arrived at independently using certain infinite series—for example, an infinite number of square root 1’s, summed and nested within each other. Either way, the answer for the Golden Ratio was the same: 1 to an irrational number, about 1.618.
The Golden Ratio first appeared in written form in the mathematician Euclid’s 300 B.C. work Elements, the “best-selling math book of all time,” said O’Connell. Even after its discovery, though, the Ratio kept cropping up in unexpected places. The Fibonacci sequence, discovered by a mathematician of the same name in the 13th Century and made up of a sequence of numbers in which each number is made up of the sum of the two before it, also has a curious connection to the Ratio: as the sequence progresses, any given number divided by the previous one grows progressively closer to the Golden Ratio.
Interestingly enough, along with the Ratio’s numerous mathematic properties, it has incredible aesthetic qualities as well. Several scientific studies, one as early as 1876, have established that the vast majority of people prefer the look of the Golden Rectangle (a rectangle proportioned according to the Golden Ratio) over rectangles of other proportions.
Not sure what the Golden Rectangle looks like? Take a look at your driver’s license or credit card. Because of its aesthetically pleasing quality, the Golden Ratio has provided a template for artists, architects and other designers throughout history. The face of the Parthenon in Greece is a Golden Rectangle, and the profile of a Volkswagen Beetle fits within a Golden Ellipse (an ellipse with Golden Ratio proportions).
Leonardo da Vinci’s masterpiece “Mona Lisa” presents perhaps the clearest evidence of the Golden Ratio’s place in art: not only is the frame itself a Golden Rectangle, but the composition of the entire portrait can be broken down into Golden Rectangles: the proportions of her face, the proportions of her nose and the measurement from her chin to her lips and the length of her nose all correspond to the Golden Ratio. “Clearly, [da Vinci] had a ruler and was measuring this stuff out,” said O’Connell.
Da Vinci’s immaculate attention to the Golden Ratio in “Mona Lisa” looks natural because it is, explained O’Connell—many proportions of the human body naturally conform to the Golden Ratio. The length of your shoulder to your wrist divided by the length from your shoulder to your elbow is close to the Golden Ratio; so is the length of your longest finger divided by the length of your pinkie. “The Golden Ratio is all throughout the human body,” said O’Connell.
Modern-day da Vincis don’t even have to do the calculations by hand, explained O’Connell; the ratio is so common in design that a template for it is built into Adobe Photoshop.
Besides occurring naturally in the human body, the Golden Ratio also appears throughout nature; “the part that I find the most impressive,” said O’Connell. Coincidentally, the Ratio happens to dictate the most efficient growth patterns for seeds and leaves, resulting in its appearance on almost any plant.
It is in the best interests of a plant growing leaves directly from a central stalk, explained O’Connell, for each to grow a certain distance radially from the one before it, in order to ensure that it does not grow directly under another leaf. Being an irrational number, the Golden Ratio turns out to be the ideal candidate—each leaf on a plant tends to grow about .618 of a rotation (around 228 degrees) around the stalk from the leaf before it.
Even as O’Connell concludes his Brown Bag semester schedule, however, he has already begun planning a presentation for the fall semester. The planned topic is online encryption technology, security that “is based on sixth grade math,” according to O’Connell. He plans on explaining how that math works—and how that even as basic as is, it’s powerful enough to keep your online information secure.
The next Brown Bag of the will be held Friday, April 10. Geology Professor Paul Belasky will be presenting “Geology: Key to Puzzles of the Ancient World.”