Geometry, Physics, and Harmonic Functions

Mathematics is a language of rigor and clarity. A plethora of symbols and words litter every student’s math textbooks, creating arguments and generalizations. Pictures, on the other hand, are more commonly left out of the texts. There are many reasons for this: how is one to visualize 5-space? How can one prove a fact with a picture? Its cliche, but things really aren’t always how they appear. Contrary to the rejection of visualization by the modern math community, Tristan Needham has produced an excellent book solely devoted to the visual side of one of the most beautiful topics in mathematics: complex analysis [3]. Another paper by the same author, [2], published in Mathematics Magazine, details a simple geometric solution to an algebraic problem. It is this paper that we will focus on. In The Geometry of Harmonic Functions, Needham takes a physical problem and its solution, generalizes it, and gives the reader multiple geometric interpretations. It is written with the assumption that the reader is familiar with complex analysis and hyperbolic geometry [1] (the latter will be briefly reviewed later). The fundamental concern of the paper is simple: how can we express values of a harmonic function in a region if we only know the values of the function on the boundary of that region? Before we can write down an equation to solve our problem, we must (as did the people who first discovered these facts) motivate the derivation of these formulae with a physical situation.