The subject of isoperimetry has a long and eventful history, both for its impact on people's imaginations and society in general and for the impetus it has given to the study of various mathematical subjects.

Isoperimetry began with the problem confronted by Queen Dido, which was to find the shape of the boundary that should be laid down (using strips of oxhide) to enclose maximum area. If one assumes a straight coastline, then the answer, which was by all appearances discovered by Queen Dido, is to lay down the hide in the shape of a semi-circle.

One finds the problem of Queen Dido colorfully described, including various embellishments of the basic problem, in the expository account that Lord Kelvin gave in 1893. If one takes account that land may vary in value, or that the coastline may be irregular, one can arrive at various more complicated problems. In a much more recent exposition, Hildebrandt and Tromba, in their book The Parsimonious Universe: Shape and Form in the Natural World (originally published as Mathematics and Optimal Form), give a much more detailed account of isoperimetric problems and their recurrence throughout history. In particular, it is interesting to see how many walled cities in the Middle Ages were constructed to have a nearly circular perimeter, or to see in general that the growth of many cities gave them a nearly circular form.

On the mathematical side, we find already in Euclid (around 300 BC) the proof that among rectangles of a given perimeter the one having the greatest area is the square. Also, various writers from antiquity speculated on optimal properties of the honeycombs of bees. When Thomas Hales proved in 2001 that regular hexagons provide a least-perimeter way to partition the plane into unit areas, it was the longest standing open problem in mathematics. As for 3D, Lord Kelvin proposed a solution consisting of relaxed, 14-sided, truncated octahedra. In 1994 D. Weaire and R. Phelan disproved Kelvin's conjecture by providing a new candidate using both 12- and 14-sided shapes.

Among the ancient Greeks who worked on the isoperimetric problem we mention Zenodorus (c. 200 - c. 140 BC) who wrote a now-lost treatise On Isoperimetric Figures and Ptolemy (c. 90 - c. 168 AD). It is thanks to Theon of Alexandria (c. 335 - c. 405 AD) who wrote a commentary on the work of Ptolemy that we know the results of Zenodorus. Al-Kindi, an Arab mathematician and the son of one of their kings, wrote in the 9th century A Treatise on Isoperimetric Figures and Isepiphanies, that is solids of given surface. There is also a lost treatise by al-Hasan ibn al-Haytham (965 - c. 1039). Abu Ja'far al-Khazin, commenting on Ptolemy's Almagest in the 10th century, generalized earlier works. Johannes de Sacrobosco (John of Holywood, c. 1195 - c. 1256 AD), an English scholar and astronomer, wrote Tractatus de Sphaera. A commentary on this treatise, dealing specifically with isoperimetry, can be found in Two New Sciences of Galileo Galilei published in 1638.

The mathematical study of the isoperimetric problem and related problems really began to take off with the advent of calculus, when people like Newton, Leibniz, the Bernoullis, and others developed systematic ways of attacking optimization problems based on the calculus, and within a few short years were attacking problems in the calculus of variations (that is, the problem of finding an optimizing path or shape of curve from among some class of curves). For example, the brachistochrone problem was formulated by Johann Bernoulli and solved by Newton and both Bernoulli brothers, Jakob (James) and Johann (John). In the same period, the problem of the shape of a hanging chain (the catenary) was posed and solved, and Newton considered the shape of projectile which would give the least air resistance (the question of designing the optimal shape for the nose-cone of a rocket or missile), but without reaching definitive conclusions. Others, including US President Thomas Jefferson, considered questions such as the optimal shape for ploughshares.

In the century following the early development of calculus by Newton, Leibniz, the Bernoulli brothers, and others, the calculus of variations was brought to a relatively advanced state, especially from the point of view of direct solutions of problems, by Euler and Lagrange. The explicit solution of the classical isoperimetric problem could be derived in those terms (using variational theory with a constraint), and many other problems could be formulated and solved. ... (click here to read the full essay)