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The Sagacity of Circles: A History of the Isoperimetric Problem - Introduction

Jennifer Wiegert

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.

Editor's Note: This article 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.

All winning papers in the HOM SIGMAA Student Contest are published in Convergence; many are also available through the HOM SIGMAA archives at

Other HOM SIGMAA Student Contest Papers in Convergence

2020: Jeffrey Powers (Grand Rapids Community College), “Did Archimedes Do Calculus?

2019: Amanda Nethington (University of Missouri – Kansas City), "Achieving Philosophical Perfection: Omar Khayyam's Successful Replacement of Euclid's Parallel Postulate."

2018: First place – Callie Lane (University of Missouri – Kansas City), "Race to Refraction: The Repeated Discovery of Snell's Law"; Second place – Christen Peters (Lee University), "The Reality of the Complex: The Discovery and Development of Imaginary Numbers," and Rachel Talmadge (University of Missouri – Kansas City), "François Viète Uses Geometry to Solve Three Problems."

2017: Co-winners – Amanda Akin (Lee University), “To Infinity and Beyond: A Historical Journey on Contemplating the Infinite,” Johann Gaebler (Harvard University), “Traditionalism: 1894 to 1925,” and Nathan Otten (University of Missouri – Kansas City), “Huygens and The Value of all Chances in Games of Fortune.”

2016: Co-winners – Brittany Anne Carlson (Salt Lake Community College), “A Latent Element of Alice’s Agency in Wonderland: Conservative Victorian Mathematics,” and William Cole (Lee University), “The Evolution of the Circle Method in Additive Prime Number Theory.”

2015: Co-winners – Samuel Patterson (University of Missouri – Kansas City), “Bernard Bolzano, a Genius Unnoticed in His Time,” and Briana Yankie (Lee University), “Examining Disproved Mathematical Ideas through the Lens of Philosophy.”

2014: First place – Jenna Miller (University of Missouri – Kansas City), "Casting Light on the Statistical Life of Florence Nightingale," and Anna Riffe (University of Missouri – Kansas City), "The Impossible Proof: An Analysis of Adrien-Marie Legendre's Attempts to Prove Euclid's Fifth Postulate"; Second place – Paul Ayers (University of Missouri – Kansas City), “Gabriel Cramer: Over 260 Years of Crushing the Unknowns," and Mary Ruff (Colorado State University – Pueblo), "Probability to 1750."

2013: Matthew Shives (Hood College), "Paradigms and Mathematics: A Creative Perspective."

2012: First place – Jesse Hamer (University of Missouri – Kansas City), “Indivisibles and the Cycloid in the Early 17th Century”; Second place – Kevin L. Wininger (Otterbein University), “On the Foundations of X-Ray Computed Tomography in Medicine: A Fundamental Review of the 'Radon transform' and a Tribute to Johann Radon.”

2011: First place – Paul Stahl (University of Missouri – Kansas City), “Kepler's Development of Mathematical Astronomy”; Second place – Sarah Costrell (Brandeis University), “Mathematics and Mathematical Thought in the Quadrivium of Isidore of Seville,” and Rick Hill (University of Missouri – Kansas City), “Thomas Harriot's Artis Analyticae Praxis and the Roots of Modern Algebra.”

2010: Co-winners – Jennifer Nielsen (University of Missouri – Kansas City), “The Heart is a Dust Board:  Abu’l Wafa Al-Buzjani, Dissection, Construction, and the Dialog Between Art and Mathematics in Medieval Islamic Culture,” Palmer Rampell (Phillips Academy and Harvard University), “The Use of Similarity in Old Babylonian Mathematics,” and Stefanie Streck (Pacific Lutheran University), “The Fermat Problem.”

2009: First place – Nathan McLaughlin (University of Montana), “The Mathematical Optics of Sir William Hamilton: Conical Refraction and Quaternions”; Second place – Tim Chalberg (Pacific Lutheran University), “Regression Analysis: A Powerful Tool and Riveting Drama”; Honorable Mention – Amy Buchmann (Chapman University), “A Brief History of Quaternions and the Theory of Holomorphic Functions of Quaternionic Variables.”

2008: First place – Mame Maloney (University of Chicago), “Constructivism: A Realistic Approach to Math?”; Second place – Woody Burchett (Georgetown College), “Thinking Inside the Box: Geometric Interpretation of Quadratic Problems in BM 13901,” and Cole McGee (Colorado State University – Pueblo), “Jean Le Rond D'Alembert: Biography of a Mathematician, Philosophe, and a Man of Letters”; Honorable mention – Mame Maloney (University of Chicago), “Pathological Functions in the 18th and 19th Centuries.”

2007: Co-winners – Rory Plante, “The Libra Astronomica and its Mathematics,” and Douglas Smith (Miami University, Ohio), “Lucas’s theorem: A Great Theorem.”

2006: Co-winners – Jennifer Wiegert, “The Sagacity of Circles: A History of the Isoperimetric Problem,” and Samantha Reynolds (University of Missouri – Kansas City), “Maria Gaetana Agnesi: Female Mathematician and Brilliant Expositor of the 18th Century.”

2005: First place – Newlyn Walkup (University of Missouri – Kansas City), “Eratosthenes and the Mystery of the Stades”; Second place – James Collingwood (Drake University), “Rigor in Analysis: From Newton to Cauchy.”

2004: Co-winners – Mark Walters, “It Appears That Four Colors Suffice: A Historical Overview of the Four-Color Theorem,” and Heath Yates (University of Missouri – Kansas City), “An Emanji Temple Tablet.”

Jennifer Wiegert, "The Sagacity of Circles: A History of the Isoperimetric Problem - Introduction," Convergence (July 2010)