A Geometric-Combinatorial Approach to the Constructing Fractals with Cubic Symmetry

   Introduction. An iterative geometric-combinatorial approach to the construction of fractals with cubic symmetry — cubofractals — is proposed, adapted for their subsequent physical implementation. A connection between the introduced cubofractals and attractors of non-stationary iterated function systems is established. Two examples of cubofractals are studied, the attractors of the corresponding iterated function systems are determined, and their Hausdorff dimensions are computed. In applied physics and materials science, fractal models can be successfully used to describe the hierarchical structure of new materials. However, perfect self-similarity at all scales is not observed in real systems. When synthesizing fractal-like structures, a minimum scale and basic structural elements that combine at higher hierarchical levels can often be identified. In this work, the authors employ this very approach to construct cubofractals.

   Materials and Methods. The proposed approach is close to constructing fractals using L-systems; however, even the simple cube arrangement rules considered by the authors require context-sensitive rules. This work establishes a connection between the considered generation algorithms and a non-stationary generalization of iterated function systems. Thus, the study of the limit transitions from prefractals to fractals is conducted within the framework of the theory of contractive mappings. The corresponding non-stationary iterated function systems are implemented in Wolfram Mathematica (the appendix provides code for the generation and visualization of cubofractals).

   Results. The adequacy and simplicity of the cubofractal generation algorithm for creating corresponding structures are demonstrated. A connection with the mathematical apparatus of non-stationary iterated function systems is established. Variants of cubofractal generation rules are studied, and different classes of the resulting structures are identified. Pointwise limits of cubofractal iterations and the limit transition in the Hausdorff metric are investigated. It is shown that by choosing a sequence a_n+1 with a given growth rate and its partitioning into Ra_n and ∆_n, we can control the iteration measure and the Hausdorff dimension of the limit. For two basic examples of cubo-fractals with sequences a_n = n∙2^n-1 + 2,  n > 0 (cubosixter) and a_{n =} 1+ 8∙5^n-2, n > 1 we calculated the fractal dimensions 1 and ln3/ln5, respectively. Backward trajectories of non-stationary systems of iterated functions are considered, and an approach to studying their attractors is proposed.

   Discussion. Cubofractals can provide a convenient approach for generating fractal structures for applications and, furthermore, they significantly enrich the theory of non-stationary iterated function systems with non-trivial examples.

   Conclusion. The limit set of the cubosister is simply the segment [0;1], while the measure of the finite iterations, which approach the segment in the sense of the Hausdorff metric, tends to zero. It can be assumed that other physical properties of the iterations (mass, strength, porosity, etc.) will also differ in the pointwise limit and in the limit in the sense of the Hausdorff metric. In this regard, it is the prefractals for the cubosister that are of research interest in applications.

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