The Sagacity of Circles: A History of the Isoperimetric Problem

Editor's Note: This article, published in June 2006, was one of the the winning articles in the 2006 competition for best history of mathematics article by a student, sponsored by the History of Mathematics SIGMAA of the Mathematical Association of America.
 

During the fourth century C.E., a Hellenistic geometer named Pappus of Alexandria introduced Book V of his Mathematical Collection not with a discussion of mathematicians past or accomplishments to follow, but rather with a preface “On the Sagacity of Bees.” By observing the near-perfect geometry of a bee’s hexagonal comb structure, Pappus attributed to the insects “a certain geometrical forethought” [Thomas, 591]. “Bees,” he wrote, “… know just this fact which is useful to them, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each” [Thomas, 593]. Pappus’s preface suggested much more than the natural efficiency of bees however. “We,” he continued, “claiming a greater share in the wisdom than the bees, will investigate a somewhat wider problem, namely that, of all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always greater, and the greatest of them all is the circle having its perimeter equal to them” [Thomas, 593].

With that, Pappus had undertaken the isoperimetric problem. Although isoperimetry contains many smaller problems within it, the central goal is to discover which of all plane figures with the same perimeter has the largest area. The question of isoperimetry was several hundred years old when Pappus addressed it in the Collection, yet even generations later, it continued to fascinate the mathematical community. Appearing in both mathematical and literary texts and captivating the minds of mathematicians even in the modern age, the isoperimetric problem serves to illustrate both the perceptiveness of ancient mathematicians and the consistency of mathematical endeavor throughout history.