May 3, 2010
Pam Pierce, professor of mathematics and computer science at Wooster, and several of her students developed an algorithm to illustrate Tarski's Circle-Squaring Problem.
WOOSTER, Ohio - Geometric puzzles have always intrigued Pam Pierce, professor of mathematics and computer science at The College of Wooster. Even as a young girl, she would spend hours working on Tangrams (dissection puzzles consisting of seven flat shapes that can be used to form various types of objects, such as ducks and swans) or trying to solve the mystery of the vanishing leprechaun on
restaurant placemats while the rest of her family was busy eating.
Pierce's fascination continues to this day as she ponders some of the more complex mysteries of geometry, including Tarski's Circle-Squaring Problem, which asks whether a circle can be broken down into a finite number of pieces and reconfigured to create a square (not to be confused with the ancient Greek's efforts to square a circle).
Building on the work of famed mathematician Miklos Laczkovich, who used the axiom of choice (which allows one to choose a representative element from each of infinitely many sets) to prove that it is possible to create a square from a circle, Pierce and several of her students in Wooster's Applied Mathematics Research Experience (AMRE) attempted to illustrate what he had done.
Using even-sided polygons, Pierce and her students developed an algorithm that could determine the number of pieces needed to transform such a polygon into a square. Like Laczkovich, their
technique requires that the pieces move by translations only - no rotations or reflections are permitted. Their research led to an integer sequence (similar to the well-known Fibonacci Sequence) that reveals how many pieces are required to transform an even-sided polygon into a square. For example, going from a four-sided polygon to a square would require just one piece, but going from a hexagon to a square would require dissection into five pieces, while going from an octagon to a square would necessitate nine pieces. As a result, their sequence begins with the numbers 1, 5, 9, 12, 15, and continues indefinitely. The numbers in this sequence are generated using a complex formula resulting from the geometry and trigonometry of the polygons and the dissection techniques developed by Pierce and her students.
"The technique used by Laczkovich is not something that we can visualize, because the 'pieces' that he uses do not have a measurable area," said Pierce. "Our goal was to approximate what he did using polygonal pieces, such as those in the tangram puzzle. Laczkovich's result is extremely impressive, and one of the goals of this research was to increase awareness of the circle-squaring problem and its solution," added Pierce, whose findings were published in the November 2009 edition of Math Horizons. "As we got further into the study of the Circle-Squaring Problem, I was pleased to find that my interest in geometric puzzles could help to shed light on this problem.
"Did we really create a square from a circle?" asked Pierce. "Not quite, but we did approximate it. As the number of sides increases, these polygons are almost indistinguishable from a circle, so to the naked eye it may appear that we have created a square from a circle. This research certainly helps us to appreciate Laczkovich's remarkable findings, and to generate some nice results concerning polygon dissections."
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