Fractal Properties of Block Elements and a New Universal Modeling Method.

Recent studies by the authors developing the theory and applications of block elements in problems of mechanics and physics have unexpectedly resulted in the possibility of interpretation of one more feature of these mechanical and mathematical objects. Following the idea of Benoit Mandelbrot on fractality, i.e., self-similarity in nature, the investigation of block elements as objects with fractal properties is carried out. It is shown that known natural phenomena, processes, and objects described by boundary-value problems for partial differential equations and sets of such equations have self-similar packed block elements as solutions. Moreover, the special role of packed block elements generated by the boundary-value problems for the simplest wave equations, or Helmholtz equations, is highlighted. In the multitude of packed block elements for the boundary-value problems of natural processes, they are objects of a discrete topological space and have the maximum topology representing all the other block elements of the multitude by their unions. At the same time, the wave equation or the Helmholtz equation are analogues of the Schrödinger equation, which is the basis of quantum mechanics. The Schrödinger equation describes the states of elementary particles in quantum mechanics. In this regard, a situation arises in which the fundamental principle of both the quantum world and the specified natural processes are self-similar named equations, which accomplish their functions on the corresponding scales. The packed block elements generated by the boundary-value problems for the same equations as for the Schrödinger equation implement their functions in a continuous medium, and the states of elementary particles are described by similar equations in quantum mechanics. [ABSTRACT FROM AUTHOR]