Math Trek

Möbius Accordion

9:27am, June 6, 2001
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Artist Susan Happersett of Jersey City, N.J., has come up with a novel twist on the venerable Möbius strip: a playful, eye-catching creation she describes as a Möbius accordion.

A Möbius strip, or band, is the remarkable one-sided surface that results from joining together the two ends of a long strip of paper after twisting one end 180 degrees. Mathematicians, magicians, artists, and many others have been playing with this intriguing object since its discovery in the 19th century by August Ferdinand Möbius (1790–1868), a professor at the University of Leipzig in Germany.

Happersett combines her interest in mathematics with a passion for visual art. In the 1980s, as an undergraduate working toward a degree in mathematics at Drew University in Madison, N.J., she found herself particularly intrigued by courses on symbolic logic and infinity. She also enjoyed an art course that involved going to New York City to visit galleries, museums, and artist's studios.

"I had always felt there were aspects of mathematics that possessed natural grace and beauty," Happersett says. "I could not understand when people told me they disliked mathematics. I became determined to find a visual way to express the intrinsic aesthetics of mathematics."

To learn the practical side of creating art, Happersett next went to Montclair State University in Upper Montclair, N.J. While earning a graduate degree in fine arts, she spent a lot of time working with grids–exploring different ways of dividing up surface areas, then plotting various functions, sequences, and series. "These graphs allowed me to study the aesthetic characteristics of functions, sequences, and series in a purely visual language," Happersett says.

Happersett followed up her art education with more mathematics, taking courses in logic at Hunter College in New York City, where she earned a master's degree in mathematics in 1993. She worked as a high school mathematics teacher until 1995.

While attending Hunter, Happersett started playing with the idea of creating her own visual language based on markings drawn in the boxes of a surface-spanning grid. "In this mathematical language, it is the number of strokes per grid space that holds significance," she says. "The placement of the strokes is random."

For several of her artworks, Happersett used the Fibonacci sequence of whole numbers to determine the number of strokes per box. In the Fibonacci sequence, each new term is the sum of the previous two terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.

Fibonacci numbers come up surprisingly often in nature. Many flowers, for example, have three, five, or eight petals. Pineapples and pine cones have rows of diamond-shaped scales, which spiral around both clockwise and counterclockwise. If you count the number of scales in one of the spirals, you are likely to find 8, 13, or 21 scales.

Happersett's set of beautifully printed, accordion-folded artbooks, Box of Growth, features grid spaces that are filled in with a specific number of strokes per row, where the number is determined by the Fibonacci sequence.

"Creating these patterns has become a type of meditation," Happersett notes. "Quite often I will make a series of drawings that are based on the evolution of a particular growth pattern. These completed drawings then work together as a book or a wall installation."

Last year, Happersett created several artworks that were displayed at a New York City artbooks show called "Fireworks." In one, she filled in the grid of squares on a strip of stiff paper, 22 inches long and 2.5 inches wide, with 13 strokes per box, using blue ink on one side and red on the other. She then creased the strip into accordion folds and glued the ends together to form a Möbius strip. Happersett named the result Thirteen Original Colonies.

The new version of Happersett's Möbius accordion features black markings on one side and white markings on the other side of a brown strip, with 13 markings, or strokes, per box. "I chose 13 because of the superstition regarding that number and the mysterious illusion created by the two faces–black and white–of the accordion," Happersett says. "Also, 13 is a Fibonacci number, so I could not resist."

Intricately printed copies of the Happersett Accordion are now available from Purgatory Pie Press in New York City. Each unfolded copy comes with a droll certificate of authenticity, along with assembly instructions and the mandatory warning that the product is a mind-boggling "Möbius device."

Happersett's approach to mathematical art focuses on algorithms. She makes the rules, then sees what happens. "By creating algorithmically generated drawings, I hope to reveal the grace and balance that I see in mathematics," Happersett says.

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Artists are not the only ones who find the Möbius strip irresistible. Authors sometimes toss the term into novels for dramatic effect–at least among readers who know what a Möbius strip is.

One of the most unlikely of such occurrences is in the amusing caper novel What's the Worst That Could Happen? by Donald E. Westlake. After describing a flamboyantly wacky hotel complex in Las Vegas, Westlake wrote the following: "When Brandon entered the spacious living room of cottage number one at three that afternoon, Earl Radburn in his knife-crease tan clothing stood at the picture window, with its view out over the Battle-Lake, at the moment peaceful, with the tall Moebius shape of the hotel beyond it."

Although this striking passage caught my eye, I have to admit that I am not at all sure how to picture a Möbius-strip hotel!

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