Image by Yoshiaki
Araki
Image by Yoshiaki
Araki
It is generally believed in the mathematical community that it is impossible today for someone without formal credentials in mathematics to engage in mathematical research or to make any contributions to mathematics. Yet there are subjects with open problems that need no accumulated mathematical arsenal to understand and to attack. It is even possible that someone innocent of training may have fresh insight that leads to fruitful results. I offer two recent examples to illustrate this phenomenon-one the well-known Dutch graphic artist, M.C. Escher, and the other an unknown San Diego homemaker, Marjorie Rice. Each tackled problems that ask which types of shapes can tile the plane, and in what manner. It is illuminating to see how each made the mathematical problems their own, asking questions in a way that made sense to them, without particularly caring how mathematicians might approach the problem. Each developed an unorthodox notation that was an essential ingredient of their methodical investigations. Each worked alone, essentially in secret, rewarded by the exhilaration of finding some answers to a large puzzle. It is interesting to contrast their questions and methods with those of mathematicians and scientists who have investigated similar questions. The professionals can learn something from the amateurs.