Introduction

When we come to explore real dimensions greater than two, matters become considerably more interesting. Indeed, in my own undergraduate studies, the question as to how to solve Laplace's equation in three or more dimensions using methods analogous to those presented here went unanswered, and remains unanswered in most undergraduate curricula the world over. I did not see the answer until my postgraduate studies, studying twistor methods in Oxford, and did not fully understand many of the geometrical aspects until my own post-doctoral work on twistor descriptions of minimal surfaces and strings. However, this author at least is convinced that many of the concepts are easily understood using only the elementary complex analysis already presented here, and this chapter is in part an attempt to get the message across in such a fashion. Another goal of this chapter and the subsequent one is to persuade some of you that, as well as being a basis for research in fundamental theoretical physics, there are also some interesting problems in basic and very applied mathematics that might well be solved with such methods, if only more people worked on it!

In some ways the presentation is easier if we make the jump straight to four dimensions, and treat the relativistic case. Results for three dimensions can then be obtained by constraining matters to a hyperplane t = 0.