Irregular oscillations and mathematical aesthetics

摘要

We have shown that a set of mathematically aesthetic principles can be written down and that these principles can be cast into a set of partial differential equations. The question is what sort of information is contained in these equations. Previously, we showed that for a certain choice of origin point data the equations collapse into a simple system that describes sinusoidal behavior along any integration path segment. For another set of origin point data, the equations collapse into a system that describes a sine within sine curve along any path segment. Here we show that for a more general set of origin point data the equations describe irregular oscillations along the coordinate axes for both the small picture and the big picture. From this we infer that the aesthetic field equations describe irregular oscillations along any path segment (at least close to the origin). The solution under study is a wave packet solution. The wave packet solution is characterized by regions of large magnitude immersed in a “vacuum” that appears empty but, in fact, has considerable structure. The wave packet solution is studied when an integration path is specified.