Symmetry

“Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.” Thus wrote Hermann Weyl, one of the great mathematicians of our time. Indeed, arguments based upon notions of symmetry are among the most powerful and elegant in mathematics. In this chapter we shall examine the role played in the study of inequalities by the simplest kind of symmetry which a plane figure can possess, namely, symmetry with respect to a line (which divides the figure into two parts, each the mirror image of the other). Symmetry strongly influenced the art of early civilizations. Its use in mathematics was begun by the Greeks. It led them to their wonderful discoveries of regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. In turn, the symmetrics of polyhedra have been partially responsible for the creation of the branch of modern mathematics known as algebraic topology. For an introduction to this point of view, I highly recommend that you read the book Geometry and the Imagination by David Hilbert and S. Cohn-Vossen, Chelsea Press, New York, 1952.

Symmetry is aesthetically pleasing, and many wonderful geometric figures can be found through constructions that involve reflections, for example, constructions of regular polyhedra (read the discussion by Hilbert and Cohn-Vossen). However, it is the abstract mathematical principle which is associated with the concept of reflection that we shall have occasion to use most often.