Professor of Mathematics Pamela B. Pierce presented the final Faculty-At-Large lecture of the year on Tuesday morning on the topic of “Circle Squaring and Other Geometric Puzzles.”† Pierce’s recent work has dealt with polygon dissection, which is slicing up two-dimensional figures and assembling them into new figures with the same area.† A basic example of this is Tangrams, a Chinese puzzle wherein one makes shapes from the pieces of a dissected rectangle.
Pierce explained that it has been proven that any two polygons with the same area are congruent by dissection, but that Max Dehn proved this was not the case for polyhedra (three-dimensional shapes) Pierce’s work focused around a question posed by Alfred Tarski: could it be done with a circle? Tarski’s challenge to the mathematical community was to take a two-dimensional circle, cut it into a finite number of pieces, and reassemble it into a square of equal area.
The answer to the problem came from Miklos Lacskovich, who proved in 1990 that it could be done, but not with scissors and paper.† Lacskovich’s proof used non-measurable sets and referred to a somewhat contentious mathematical concept called the axiom of choice.† Pierce spent some time describing the axiom of Choice, which is a concept in set theory that allows mathematicians and logicians to choose one item from each set even in situations with an infinite number of sets and no rule for deciding which item to choose.
Using this axiom, Lacskovich proved not only that Tarski’s problem could be solved, but that it could be solved using only translations (that is, without reflecting or rotating the pieces) and that the upper limit for the number of pieces required was on the order of 10 to the 50th.
Pierce’s research, which included a number of students at the College, picked up where Lacskovich’s work left off.† Pierce focused on the fact that, while strong theoretically, Lacskovich’s work was hard to understand and nearly impossible to visualize.
She and her students returned to polygons and focused on approximating Lacskovich’s result using measurable pieces.† They created a formula to calculate the number of measurable pieces required to dissect a polygon with 2n sides and reconstitute it into a square, then used computer graphics to illustrate the process, showing the particular cuts and pieces for any given polygon.
The formula they devised yielded a new sequence of integers, namely the minimum number of pieces for each different kind of polygon, and the group registered this sequence on an online database, in case those numbers some day correspond to another integer series.
Ultimately, Pierce admitted that there was no solution to Tarski’s challenge but the one Lacskovich presented. “To truly create a square from a circle,” she said, “we need to trade in the scissors for the axiom of choice.”