New book makes geometry swing and twist

October 02, 2002

WEST LAFAYETTE, Ind. -- It's not a nostalgic return to the '50s and '60s, but a Purdue University professor's new book still may have readers doing the "twist."

Greg Frederickson's book, "Hinged Dissections: Swinging and Twisting," delves into a world in which triangles can be transformed into squares, crosses into hexagons, and back again -- all with the grace of dancers swinging around a ballroom floor.

True to its subtitle, Frederickson's book illustrates how geometric figures can be cut into pieces, which are then attached to one another by hinges that allow the pieces to swing into new positions and form a different figure. Mathematical puzzles like these were popular in newspapers around the turn of the 20th century. Although a few of these dissections were known to work with hinges that have a swinging motion, Frederickson discovered enough new hinged dissections to fill a whole book.

Yet this professor of computer sciences (and, incidentally, a child of the 60s) added a new dimension by including the idea of "twisting."

"Instead of simply swinging pieces on a flat surface, I discovered a new kind of transformation that relies on twisting around the pieces in three dimensions," Frederickson said. "It boggled my mind when I realized that I could twist around a set of pieces to form, for instance, either a square or an equilateral triangle."

"Hinged Dissections: Swinging and Twisting" was published by Cambridge University Press last month at a price of $45. "Dissections: Plane and Fancy," Frederickson's earlier book on unhinged dissections, appeared in 1997.

With more than 500 illustrations and whimsical prose suggesting a vivid oral presentation, Frederickson takes the reader on a remarkable journey from the simplest of geometric puzzles to complex transformations involving solid objects. Along the way, the author highlights the colorful history of the field and explains the methods with which he has created hinged dissections. One method uses tessellations, which are repeating patterns formed from identical tiles.

"Readers may be familiar with tessellations from some of the lovely work of the artist M.C. Escher," Frederickson said. "My book takes off in a different but equally fascinating direction of using the symmetry of the tessellations to create hinged dissections."

Frederickson, whose specialty is designing computer algorithms that use a minimum of time and memory, found that the emphasis on efficiency in his research spilled over into his passion as he tried to minimize the number of pieces in his transformed shapes. He is equally motivated by a fundamental curiosity about how geometry and motion interact.

"These issues are also addressed by researchers who think about packing objects compactly into space launch vehicles," he said. "Once the object has been transported up into space and is ready to be deployed, it may need to be unfurled or otherwise transformed into a desired configuration, using hinged motion."

Aside from the possible applications, however, Frederickson swings figures -- and writes books about this unusual activity -- for a much simpler reason.

"It's great fun," he said. "Try it, and you'll find yourself swinging, too."
Writer: Chad Boutin, (765) 494-2081,
Source: Greg N. Frederickson, (765) 494-6016,

Related Web sites:
Book Web site:
Movies of swinging pieces:
Photos of a square/triangular table made from swinging pieces:

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