'Dissections' book about
more than just geometry
By Kevin Cullen, Journal and Courier
Turn a star into a pentagon.
Cut an octagon into five pieces and make them into a square.
If you can do that, or want to, you'll enjoy a unusual new book by Purdue University computer science professor Greg N. Frederickson.
It's called Dissections: Plane and Fancy. The 310-page work is devoted to the challenges and history of geometric dissections; that is, the mathematical art of making a new figure from the smallest number of parts of a dissected figure.
"It's a book that you can enjoy on several levels," Frederickson said in an interview in his book-filled office.
Some readers will be fascinated by the ingenious, mind-bending techniques used. Others will focus on the colorful origins of dissections. Artists will appreciate the wonderful symmetry of the geometry itself.
"If you enjoy woodworking and understand some high school geometry, you can build many of the puzzles in this book," Frederickson wrote. And "you need not make a trip to the hardware store."
The book was fun to write because curiosity and chance drew him into unexplored regions, he said.
"I had no plan. I fell into it, and was amazed to find how much had already been done, but was never published, and how much one could still do," he said.
A thorough history of dissections and mini-biographies of its chief devotees through the centuries had never been compiled before.
"Plato talks about cutting two equal squares into pieces to form a large square," Frederickson said. "When Western civilization died down, the Arabs took over. A number of (dissections) are in math texts from 900 to 1000 A.D."
Frederickson explains solution methods, assuming only a basic knowledge of high school geometry, then poses puzzles to solve. He also introduces the people who have worked on such problems, traveling from the palace school of 10th century Baghdad to the mathematical puzzle columns in turn-of-the-century newspapers.
The advance of literacy during the 19th century led to a hunger for dissection problems. Many written by English puzzlist Henry Dudeney and American Sam Loyd appeared in newspapers, popular magazines, and advertising giveaways.
Rival publications even sponsored competitions and offered prizes.
"As I tracked down the famous, the not-so-famous and the obscure, my curiosity grew to fascination, and my fascination to near-obsession," Frederickson wrote.
Through an interlibrary loan, he found many long-forgotten articles on the subject and was able to dig out earlier references and make connections that other authors had missed. He also corresponded with dissection buffs worldwide, allowing their creations to appear in book form for the first time.
A discussion of each problem highlights the techniques and their underlying mathematical properties and is accompanied by numerous diagrams.
Dissection puzzles teach lessons about the mathematical concept of area. Shapes can be cut apart and rearranged, but the total area does not change, he said.
Frederickson became interested as a high school student when his parents bought him a book on the subject by Harry Lindgren.
After graduating in economics from Harvard University, Frederickson taught junior high math for several years and used dissections in displays.
"At some point, I figured out how to improve one of the dissections by finding a solution with one fewer piece," he said. As he played with the puzzles more, he found more improvements.
Frederickson began corresponding with Lindgren. In 1972, Frederickson published an appendix to Lindgren's original book. Soon after that, he embarked on his career in academia and set aside his interest in dissections.
In 1991, he received a letter from Martin Gardner, who was revising a book on mathematical games. For many years, Gardner wrote a recreational mathematics column for Scientific American magazine.
Gardner asked what was new in dissections. That led Frederickson to "pull all of this information out of the closet and begin thinking about them again."
He began writing the book in 1994. Gardner provided letters and names of others in the field who offered Frederickson valuable information and unpublished material.
Frederickson also visited the Library of Congress, Smithsonian Institution, Chicago Historical Society and other repositories.
Gardner calls dissections "one of the most elegant and surprising branches of recreational mathematics. No one knows more about this, or is more skilled in breaking old records, than Greg Frederickson. His book will be a classic."
Though most of the book is based on high school-level math, Frederickson used simple but precise graphics so that readers of all levels can skim through even the most difficult sections.
Computer software now allows geometric figures to be formed, divided into precise color-coded sections, then moved, rotated, and reassembled. Others, however, still stick to the traditional compasses and straight edges to do their work.
"This was so much fun," Frederickson said. "So unexpected and surprising ... What's nice about all this is that the more you get into it, the more opportunities you have to look at nice things and embellish them."