Concept of ubiquitiform in applied nonlinear science

A fundamental idea on ubiquitiformal (a terminology coined here for a finite order self-similar or self-affine structure) geometry in natural world was proposed. That is, any physical object in nature should be ubiquitiformal rather than fractal. It is shown mathematically that a ubiquitiform must be of integral dimension. It is also demonstrated that the dimension of the initial element on which a fractal is constructed changes abruptly, which results in the divergence or singularity of the integral dimensional measure of the fractal, thus using a fractal to approximately describe a ubiquitiform is unreasonable. Moreover, it is also shown that some inherent difficulties of fractal application in nonlinear science can be avoided by introducing the concept of ubiquitiform.