Balancing on the edge, the golden ratio, the Fibonacci sequence and their generalization

Abstract

The golden ratio and Fibonacci numbers are found to occur in various aspects of nature. We discuss the occurrence of this ratio in an interesting physical problem concerning center of masses in two dimensions. The result is shown to be independent of the particular shape of the object. The approach taken extends naturally to higher dimensions. This leads to ratios similar to the golden ratio and generalization of the Fibonacci sequence. The hierarchy of these ratios with dimension and the limit as the dimension tends to infinity is discussed using the physical problem.