Steiner Symmetrization

Abstract

The Swiss geometer Jakob Steiner (1796 – 1863) devoted much of his energy to seeking simple principles from which theorems in geometry could be derived in a natural way. The Isoperimetric Theorem provided one venue for this process and symmetrization was the simple principle he discovered. Mathematical mythology attributes the classical Isoperimetric Theorem (stating that the circle is the plane figure enclosing the greatest area for a given perimeter or equivalently the figure of least perimeter enclosing a given area) to Queen Dido, founder of Carthage. More reliable authority (see Knorr [1] and [2]) credits Zenodorus with a proof in the second or third century B.C. Steiner himself refers to another Swiss mathematician Simon-Antoine-Jean L’Huilier (1750 – 1840) as the one who provided the first definitive exposition of the Isoperimetric Theorem. L’Huilier published a treatise on the subject (in Latin) in 1782. Though an admirer of L’Huilier, Steiner was not satisfied with the status of the Isoperimetric Theorem, for Steiner felt that while L’Huilier’s work was often cited, L’Huilier’s methods were not rightfully appreciated or understood. Steiner thus sought a deeper and clearer understanding of the classical Isoperimetric Theorem and its variants.