Mathematical Modeling and Recursive Algorithms for Constructing Complex Fractal Patterns

Abstract

In this paper, we present mathematical geometric models and recursive algorithms to generate and design complex patterns using fractal structures. By applying analytical, iterative methods, iterative function systems (IFS), and L-systems to create geometric models of complicated fractals, we developed fractal construction models, visualization tools, and fractal measurement approaches. We introduced a novel recursive fractal modeling (RFM) method designed to generate intricate fractal patterns with enhanced control over symmetry, scaling, and self-similarity. The RFM method builds upon traditional fractal generation techniques but introduces adaptive recursion and symmetry-preserving transformations to produce fractals with applications in domains such as medical imaging, textile design, and digital art. Our approach differs from existing methods like Barnsley’s IFS and Jacquin’s fractal coding by offering faster convergence, higher precision, and increased flexibility in pattern customization. We used the RFM method to create a mathematical model of fractal objects that allowed for the viewing of polygonal, Koch curves, Cayley trees, Serpin curves, Cantor set, star shapes, circulars, intersecting circles, and tree-shaped fractals. Using the proposed models, the fractal dimensions of these shapes were found, which made it possible to create complex fractal patterns using a wide variety of complicated geometric shapes. Moreover, we created a software tool that automates the visualization of fractal structures. This tool may be used for a variety of applications, including the ornamentation of building items, interior and exterior design, and pattern construction in the textile industry.

1. Introduction

Fractals are self-similar structures that exhibit complex geometric properties at various scales. Unlike traditional Euclidean shapes, fractals possess fractional dimensions, making them invaluable for modeling natural and artificial patterns that exhibit irregularity. Since their introduction by Benoît Mandelbrot, fractals have been extensively studied for their ability to capture intricate structures found in nature, such as coastlines, river networks, cloud formations, tree branches, and biological tissues. These properties allow fractals to be used in scientific modeling, computational simulations, and artificial intelligence applications. Mathematically, fractals follow recursive generation rules, which means a simple initial structure is transformed through repeated operations. Various fractal generation techniques have been developed over time, including iterated function systems (IFS), Lindenmayer systems (L-systems), escape-time fractals, and random fractal processes. Despite their mathematical elegance and practical utility, traditional fractal generation techniques often lack flexibility in parameter adjustments, symmetry control, and computational efficiency, limiting their ability to model highly detailed structures in a computationally feasible manner [1,2].

Fractals exhibit three core mathematical properties: self-similarity, fractional dimensionality, and recursive generation. Self-similarity means that a fractal pattern appears identical at different scales. This property is frequently observed in natural formations, including blood vessels, ferns, and coastlines. Fractional dimensionality is another key characteristic of fractals, where their Hausdorff dimension is greater than their topological dimension. This property allows fractals to model objects that partially fill space, unlike classical Euclidean structures. Fractal structures can be generated through different iterative techniques. Generator iteration involves replacing specific parts of a shape with a modified version of itself, as seen in Koch curves and Sierpiński triangles. IFS-based fractals use affine transformations, such as scaling, rotation, and translation, to generate self-similar structures. Formula-based fractals, including the Mandelbrot set and Julia sets, employ complex number mappings to create visually intricate patterns. These iterative methods form the foundation of modern fractal generation algorithms, which are applied in pattern recognition, scientific simulations, and artistic design [3,4,5].

Fractal geometry has numerous applications across multiple fields:

• medical imaging and biomedical analysis: used in cancer detection, heart rate variability analysis, and lung disease modeling through fractal dimension analysis;
• scientific computing and engineering: applied in fluid dynamics, material analysis, and aerodynamics for complex shape modeling;
• artificial intelligence and pattern recognition: utilized in data compression, facial recognition, and feature extraction in machine learning;
• financial markets and econophysics: used to analyze stock market trends and financial time series;
• digital art and textile design: used for generative artwork, fabric pattern creation, and digital media applications.

These diverse applications demonstrate the significance of fractals in both theoretical research and real-world problem solving.

Despite their wide applications, traditional fractal generation methods suffer from several limitations. Many classical approaches rely on predefined transformation rules, reducing their ability to adapt to custom design requirements. Recursive algorithms often require high computational power, leading to long processing times, which is especially problematic in real-time applications. Additionally, structural distortions occur when scaling complex fractal patterns, affecting their symmetry and aesthetic consistency. Several studies have attempted to address these challenges by introducing hybrid fractal generation methods, optimized recursive algorithms, and improved transformation techniques. However, these approaches still lack fine-grained control over fractal symmetry, adaptive recursion depth, and computational efficiency. There is a growing need for an enhanced fractal modeling technique that balances complexity, efficiency, and flexibility.

In this work, we introduce the recursive fractal modeling (RFM) method, a novel approach designed to address these limitations by providing enhanced control over fractal properties, including symmetry, scaling, and self-similarity. Unlike standard methods, the RFM method incorporates adaptive recursion and symmetry-preserving transformations, enabling the generation of intricate patterns with consistent structural properties. Our method also optimizes computational efficiency by minimizing redundant calculations through in-place recursion and intermediate result caching, achieving faster convergence with fewer iterations. This study builds upon foundational methods in fractal geometry, such as iterated function systems (IFS), contraction mappings, and recursive algorithms, which are essential for creating complex fractal patterns. These concepts closely align with the theories outlined in Fractals Everywhere by Michael Barnsley, a foundational work that has significantly influenced the study and application of fractal mathematics [6]. By incorporating these established methods, we ensure that our models adhere to widely recognized principles in fractal generation. Our key contributions include:

• development of a novel recursive algorithm—the RFM method integrates adaptive recursion, allowing for flexible scaling and depth control while maintaining structural consistency;
• implementation of symmetry-preserving transformations—ensuring that fractal patterns maintain a high degree of visual and geometric balance, making them suitable for high-precision applications;
• optimization of computational efficiency—reducing processing time through in-place recursion and intermediate result caching, leading to faster convergence with fewer iterations;
• application of the RFM method in real-world scenarios—demonstrating the effectiveness of our approach in fields such as digital art, and textile design.

Related Works

Fractal geometry has proven to be highly beneficial in fashion and textile engineering, particularly in automated pattern generation, fabric design, and decorative aesthetics. Several studies have investigated how recursive fractal algorithms can be applied to fashion textiles, woven patterns, and computerized fabric design.

Ge et al. [7] explored the aesthetic impact of fractal graphics in textile design, demonstrating their ability to create visually dynamic and mathematically structured clothing patterns. Their study revealed how fractal algorithms could generate unique textile prints, improving both design diversity and production automation. Long and Lu [8,9] extended this research by integrating fractal pattern generation into jacquard fabric manufacturing, analyzing how factors like color, texture, and weave structure influence the final textile product. Further applications of fractals in textiles involve L-systems, which allow for the generation of recursive geometric structures [10,11,12,13,14,15,16]. Kejun Cen et al. [17] utilized L-system-based fractal modeling to create woven textile designs with high levels of self-similarity, improving fabric symmetry and repeatability. Wang et al. [14] studied the use of J-set visualization techniques in designing fashion and decorative patterns, employing time-lapse fractal algorithms to generate complex motifs.

The computational and mathematical principles of fractal generation have been extensively explored in research. Negi et al. [18,19,20,21] presented multiple recursive algorithms for fractal construction, including the “Chasing Time Algorithm”, “Random Repetition Algorithm”, and “Deterministic Recursive Method”. These algorithms have been instrumental in improving the computational efficiency of fractal rendering, reducing the time complexity associated with recursive calculations. Edgar et al. [22] introduced set-theoretic approaches to fractal generation, offering a comprehensive method for visualizing stochastic, algebraic, and geometric fractals. Their study justified the application of fractal models in scientific simulations, topology, and spatial modeling. Gdaweic et al. [23] further extended this research by developing root-finding algorithms for generating highly detailed fractal structures, optimizing the accuracy and convergence speed of fractal-based systems.

Beyond their scientific applications, fractals have been extensively studied in the literature on computational creativity and theoretical physics. Liu et al. [24] explored how fractals contribute to nonlinear systems theory, emphasizing their significance in chaos modeling, wave dynamics, and complex systems simulations. Mohsen et al. [25] analyzed natural fractal formations, developing mathematical models that replicate organic patterns like leaf venation, snowflake structures, and coastline formations. In the field of generative art and procedural content creation, Ahammad et al. [26] introduced novel algorithms for Mandelbrot and Julia set visualization, influencing advancements in digital art, animation, and special effects. Gdaweic et al. [27,28] expanded on these ideas by incorporating multi-parameter recursive transformations to create color-mapped fractal images, showcasing how fractals enhance artistic and computational creativity.

The ability of fractals to model biological growth patterns, vascular networks, and tissue structures has made them highly relevant in medical imaging and biological research. Losa et al. [2] examined how fractal dimension analysis can be used to detect and classify tumors, revealing that malignant growth follows fractal-like expansion patterns. This finding has been instrumental in early cancer detection, automated image segmentation, and tumor classification. Chen et al. [3] applied fractal segmentation techniques to lung X-ray and MRI images, demonstrating that fractal-based approaches significantly improve disease detection and classification accuracy. Their research focused on quantifying the self-similarity of lung structures and optimizing medical image processing through fractal-based artificial intelligence models.

Fractal analysis has been widely adopted in engineering, fluid dynamics, and material science. Xie et al. [4] enhanced the Kozeny–Karman equation using fractal-based permeability models, improving conductivity calculations in woven textiles and porous materials. Their research demonstrated that fractal dimensions provide a more accurate representation of gas and fluid flow behavior in engineered materials. Xiong et al. [5] expanded on this concept by applying fractal-based fitting models to gas permeability analysis, showing that fractal dimensions can be used to predict airflow through structured materials with improved precision. Sheyko [29] further contributed to the engineering applications of fractals by formulating equations for geometric fractals, such as the Serpinski carpet, Menger sponge, and Star of David fractals, offering new approaches for architectural and materials engineering design.

This review underscores the widespread applicability of fractals in mathematics, computational science, and creative fields. Despite their extensive use, existing fractal generation methods lack scalability, structural consistency, and computational efficiency. The recursive fractal modeling (RFM) method presented in this study addresses these challenges, demonstrating higher adaptability, improved precision, and enhanced real-world applicability. Through rigorous experimental validation, the RFM method establishes itself as a novel and efficient approach to fractal modeling, expanding its potential use in scientific research, engineering, AI, and digital media.

2. Development of Mathematical and Geometric Models of Complex Fractal Structured Objects

Our approach leverages and extends iterative function systems (IFS), L-systems, and R-functions with tailored modifications that enhance their ability to create intricate, high-resolution fractal patterns suited for industrial applications. While these methods are commonly used for fractal generation, we introduced specific recursive adaptations that allow for the production of highly complex, culturally significant patterns, which are particularly useful in textile design, architectural decor, and digital art. The enhancements applied to IFS involve the integration of variable scaling factors and unique recursive transformations that control the arrangement and density of nested structures. This allows us to generate fractals with greater symmetry and precision in spatial distribution, which is critical for patterns requiring a high degree of visual regularity and complexity, such as traditional star, circular, and polygonal designs.

Similarly, our adaptation of the R-function method includes adjustments in the boundary conditions and recursion parameters, which enable us to model intersecting circular and tree-like fractals with an intricate level of detail. These adaptations result in geometric configurations that maintain structural integrity across scales, enabling the creation of fractal models that can seamlessly expand or contract for various design applications without loss of detail. In addition to these mathematical modifications, we developed a visualization algorithm specifically designed to automate the fractal generation process. This tool is optimized for high-resolution pattern creation, facilitating the efficient design of complex fractal structures for industrial applications, such as fabric printing, ceramic and porcelain decor, and architectural patterning. The software also supports customization for user-defined parameters, further enhancing its utility in pattern design and visualization.

2.1. Geometric Modeling Using Analytical Methods

The analytically developed mathematical model of fractals, consisting of regular polygons, is as follows:

First of all, the main geometric elements of a regular polygon are selected. That is, M(x₀, y₀) are the coordinates of the center of the circle drawn outside the regular polygon. R is the radius of the circle drawn outside the regular polygon, the increase (or decrease) of the L-radius. a is the starting angle; n is the number of sides. S is the number of recursions (or steps).

Step 1:

$$\begin{array}{c}S=1\\ x_1=x_0+R\mathit{cos}\left(\alpha +{\displaystyle \frac{360}{n}}\right); y_1=y_0+R\mathit{sin}\left(\alpha +{\displaystyle \frac{360}{n}}\right);\\ M\left(x_1,y_1\right),\\ M\left(x_0+R\mathit{cos}\left(\alpha +{\displaystyle \frac{360}{n}}\right), y_0+R\mathit{sin}\left(\alpha +{\displaystyle \frac{360}{n}}\right)\right),\\ {\omega }_1=f_0\left(x_0,y_0\right)\cup f_1\left(x_1,y_1\right)\end{array}$$

Step 2:

$$\begin{array}{c}S=2\\ x_1=x_0+R\mathit{cos}\left(\alpha +{\displaystyle \frac{360}{n}}\right);\\ y_1=y_0+R\mathit{sin}\left(\alpha +{\displaystyle \frac{360}{n}}\right);\\ F=M_1\left(x_1,y_1\right),\\ F=M_1\left(x_0+R\mathit{cos}\left(\alpha +{\displaystyle \frac{360}{n}}\right),y_0+R\mathit{sin}\left(\alpha +{\displaystyle \frac{360}{n}}\right)\right).\\ x_2=x_0+R\mathit{cos}\left(\alpha +{\displaystyle \frac{720}{n}}\right);\\ y_2=y_0+R\mathit{sin}\left(\alpha +{\displaystyle \frac{720}{n}}\right);\\ F=M_1\left(x_0+R\mathit{cos}\left(\alpha +{\displaystyle \frac{360}{n}}\right),y_0+R\mathit{sin}\left(\alpha +{\displaystyle \frac{360}{n}}\right)\right)+M_2\left(x_0+R\mathit{cos}\left(\alpha +{\displaystyle \frac{720}{n}}\right),y_0+R\mathit{sin}\left(\alpha +{\displaystyle \frac{720}{n}}\right)\right),\\ {\omega }_2={\omega }_1\cup f_2\left(x_2,y_2\right).\end{array}$$

At step s = i, the formula looks like this:

$$\begin{array}{c}FR={\displaystyle \sum _{j=1}^n}\left({\displaystyle \sum _{i=1}^n}{M}_i^j\left(x_0+R\mathit{cos}\left(\alpha +{\displaystyle \frac{i\times 360}{n}}\right)\right),\left(y_0+R\mathit{sin}\left(\alpha +{\displaystyle \frac{i\times 360}{n}}\right)\right)\right),\\ {\omega }_n={\omega }_{n-1}\cup f_n\left(x_n,y_n\right).\\ S=S-1;\hspace{1em}R={\displaystyle \frac{R}{L}}.\end{array}$$

It is known that nonlinear dynamical systems have several steady states. After several iterations, the state of the dynamic system depends on its initial state. If the two-dimensional space is phased, it is possible to obtain color phase images of these systems by coloring the field of gravity points with different colors. Fractal images can be obtained by changing the color selection algorithm.

It is possible to create very complex non-trivial structures using mathematical primitive algorithms. The results obtained at different iteration steps are presented in Figure 1.

Analytically, when creating fractals consisting of circles, the main geometric elements are first selected. That is, M(x₀, y₀) are the coordinates of the center of the circle. R is the radius of the circle, with enlargement (or reduction) of the radius. L is the initial angle; n is the number of elements. S is the number of recursions (or steps).

Then, the general formula looks like this:

$$\begin{array}{c}FR={\sum }_{j=1}^n\left({\sum }_{i=1}^n{M}_i^j\left(x_0-\left(R\cdot {\displaystyle \frac{\left(l-1\right)}{l}}\right)\cdot \mathit{cos}\left({\displaystyle \frac{\left(\left(\alpha +n\cdot \left(\frac{360}{k}\right)\right)\cdot \pi \right)}{180}}\right),y_0-\left(R\cdot {\displaystyle \frac{\left(l-1\right)}{l}}\right)\cdot \mathit{sin}\left({\displaystyle \frac{\left(\left(\alpha +i\cdot \left(\frac{360}{k}\right)\right)\cdot \pi \right)}{180}}\right)\right)\right)\\ R={\displaystyle \frac{R}{L}}.\end{array}$$

(1)

As a result, we achieve circular fractals at different iteration steps, as shown in Figure 2.

2.2. Geometrical Modeling Using L-Systems and IFS Methods

Initially, L-systems were introduced in formal language learning and used in biological selection models. It was found that many self-similar fractals can be constructed using them, including the Koch snowflake and the Serpin carpet, while some other classical constructions, such as Peano curves (Peano, Hilbert, and Serpin napkin), also fit this scheme. And, of course, L-systems open the way to the construction of an infinite variety of new fractals.

For the graphical implementation of L-systems, the term turtle graphics (Turtle) is used. In this case, the point (Turtle) moves across the screen with special steps, as a rule, looking for its own track, and it can also move without being drawn. We have three parameters at our disposal, which are the coordinates of the turtle and the angle of the direction that it can turn [30].

To obtain the result of the L-system, we introduce the following notations:

F—draw a trace and move one step forward.

b—take a step forward without leaving a trace.

[—open field for entering commands and variables.

]—close the field for entering commands and variables.

+—increase the angle α by θ (clockwise).

−—reduce the angle α by the amount θ (counterclockwise).

The L-system, corresponding to the Koch snowflake, is defined as the following:

The starting word generates rules in parallel to the axiom:

axiom = W0 = W0 (F, b, +, −, [ , ]).

The process is then repeated to obtain a sequence of words W0, W1, W2, … with a complex internal structure of characters. For example, by using the axiom F and the rule F+F−F−F+F, we can generate the sequence W0, W1, W2, …. (See Figure 3).

The words W0, W1, and W2 are shown as a turtle footprint (where θ = π/2, the length scale in all images is different).

One of the many ways to describe affine transformations for repeated function systems (IFS) is as follows:

$$\begin{array}{c}{x}^{\prime }=x·r·cos\left(\theta \right)+y·s·sin\left(\pi \right)+h;\\ y^{\prime }=-x·r·sin\left(\theta \right)+y·s·cos\left(\pi \right)+k.\end{array}$$

There is a way to obtain the resulting shape after an infinite number of iterations.

$$\begin{array}{c}{x}_{n+1}\leftarrow f\left[x_n,y_n\right];\\ y_{n+1}\leftarrow f\left[x_n,y_n\right].\\ x_{n+1}=a{x}_n+b{y}_n+c,\\ y_{n+1}=d{x}_n+e{y}_n+f.\end{array}$$

For any given image, there are four different sets of values for (a, b, c, d, e, f) in the interval [−1, 1]. The following relations must be fulfilled between these coefficients:

$$\begin{array}{c}{a}^2+d^2<1,\\ b^2+e^2<1,\\ a^2+b^2+d^2+e^2<1+{\left(ae-db\right)}^2.\end{array}$$

The geometric model for constructing pentagonal fractals using the IFS method is as follows. Five equivalent equilateral triangles are formed by connecting the center of the pentagon to each vertex with straight lines. Each of the central angles is 360°/5 = 72°. If the size of the remaining two angles in the equilateral triangles is A, then the expression A + A + 72 = 180 is reasonable, and it follows that the size of each external angle is also equal to 72° (see Figure 4 and Figure 5).

Consider the original pentagon P(0) with bottom vertices at (0, 0) and (1, 0). The Figure 5 shows two of the scaled pentagons of P(1) along the x-axis. These pentagons form an equilateral triangle with a base angle of 72°. So, the length of the base is equal to $2r\mathit{cos}(72°)$. Using the fact that the total length of the base of pentagons and triangles is the same as the length of the base P(0), we perform the following calculations:

$$\begin{array}{c}2r+2r\mathit{cos}72°=1,\\ r={\displaystyle \frac{1}{2+2\mathit{cos}72°}}={\displaystyle \frac{1}{2\left(1+\frac{\sqrt{5}-1}{4}\right)}}={\displaystyle \frac{2}{3+\sqrt{5}}}\end{array}$$

The translation of the second pentagon is shifted to the right by 1 − r = 0.618.

Vectors are used to translate each of the pentagons to a different scale. For example, in the Figure 5, the three vectors drawn in black end up at the third lower-left corner of the pentagon, showing how they are translated. Using the angles in Figure 6 and Figure 7, this expression is derived:

$$(r,0)+(r\mathit{cos}72°, r\mathit{sin}72°)+(r\mathit{cos}36°, r\mathit{sin}36°)=(0.809,0.588)$$

We continue the process for the remaining two pentagon broadcasts:

$$\begin{array}{c}(-r\mathit{cos}72°, r\mathit{sin}72°)+(r\mathit{cos}36°, r\mathit{sin}36°)+(r\mathit{cos}72°, r\mathit{sin}72°)=(0.309, 0.951)\\ (-r\mathit{cos}72°, r\mathit{sin}72°)+(r\mathit{cos}36°, r\mathit{sin}36°)+(-r, 0)=(-0.191, 0.588)\end{array}$$

To make this an iterative system, the image starts from the original polygon and is scaled by a factor of r = 0.382. In the next step, the six copies are rotated to fit inside the original polygon. The middle pentagon must first be rotated 180°. Now, the same process is repeated for each of the six polygons to attain the second iteration and continue indefinitely. The recursive functions for this process are generated as follows:

$$\begin{array}{c}{F}_1=f_1\left(F_0\right)\cup f_2\left(F_0\right)\cup f_3\left(F_0\right)\cup f_4\left(F_0\right)\cup f_5\left(F_0\right)\cup f_6\left(F_0\right),\\ F_2=f_1\left(F_1\right)\cup f_2\left(F_1\right)\cup f_3\left(F_1\right)\cup f_4\left(F_1\right)\cup f_5\left(F_1\right)\cup f_6\left(F_1\right),\\ \dots \\ F_n=f_1\left(F_{n-1}\right)\cup f_2\left(F_{n-1}\right)\cup f_3\left(F_{n-1}\right)\cup f_4\left(F_{n-1}\right)\cup f_5\left(F_{n-1}\right)\cup f_6\left(F_{n-1}\right).\end{array}$$

To generate these polygonal fractals using the IFS method, six functions are constructed using the scale factor:

$$\begin{array}{c}r={\displaystyle \frac{3-\sqrt{5}}{2}}=0.381966.\\ f_1(x)=\left[\begin{array}{cc}0.382& 0\\ 0& 0.382\end{array}\right]x,\\ f_2(x)=\left[\begin{array}{cc}0.382& 0\\ 0& 0.382\end{array}\right]x+\left[\begin{array}{c}0.618\\ 0\end{array}\right],\\ f_3(x)=\left[\begin{array}{cc}0.382& 0\\ 0& 0.382\end{array}\right]x+\left[\begin{array}{c}0.809\\ 0.588\end{array}\right],\\ f_4(x)=\left[\begin{array}{cc}0.382& 0\\ 0& 0.382\end{array}\right]x+\left[\begin{array}{c}0.309\\ 0.951\end{array}\right],\\ f_5(x)=\left[\begin{array}{cc}0.382& 0\\ 0& 0.382\end{array}\right]x+\left[\begin{array}{c}-0.191\\ 0.588\end{array}\right],\\ f_6(x)=\left[\begin{array}{cc}-0.382& 0\\ 0& -0.382\end{array}\right]x+\left[\begin{array}{c}0.691\\ 0.951\end{array}\right].\end{array}$$

The dimension of the fractal generated in step 4 of the iteration, consisting of six non-overlapping pentagons, is calculated as follows:

$${\displaystyle \sum _{k=1}^6}{r}^d=1\hspace{0.33em}\Rightarrow d={\displaystyle \frac{\mathit{log}\left(\frac{1}{6}\right)}{\mathit{log}(r)}}={\displaystyle \frac{\mathit{log}(6)}{\mathit{log}\left(\frac{2}{3-\sqrt{5}}\right)}}=1.86172.$$

For the fractal image obtained at step i = 4 of Figure 8, the fractal dimension is d = 1.86172.

2.3. Mathematical Modeling Using RFM Methods

The RFM method of V. L. Rvachev can be used to write analytical equations of geometric shapes in the field of fractals. This method is a numeric function with a real variable whose pointers are completely defined by the pointers of the arguments in the corresponding parts of the intervals of the axis. The main concepts of the RFM method are that the signs of the function in each interval are coordinated with the signs of the arguments. This method is effective for the mathematical determination of fractal shapes and the description of their geometry.

If there exists such a following logical function F and its arguments sign(z) = F(sign(x), sign(y)), the numerical function z = z(x, y) is called an R-function.

Each RFM corresponds to a single basis-logic function. The set of R-functions is closed in terms of the overlapping of elements. If the set of all overlapping elements of N has a non-empty intersection with each branch of the set of R-functions, then the system of R-functions is considered sufficiently complete for N. This concept is important in fractal geometry and is used to determine analytical equations of geometric shapes. The system of R-functions is compatible with each part of a given set, ensuring the quality and accuracy of its performance. The most widely used R-function complete system (−1 < α ≤ 1) at R_α is following:

$$R_a=\left\{\begin{array}{l}x{\wedge }_{\alpha }y\equiv {\displaystyle \frac{1}{1+\alpha }}\left(x+y-\sqrt{x^2+y^2-2\alpha xy}\right),\\ x{\vee }_{\alpha }y\equiv {\displaystyle \frac{1}{1+\alpha }}\left(x+y+\sqrt{x^2+y^2-2\alpha xy}\right),\\ \bar{x}\equiv -x.\end{array}\right.$$

At α = 0, we have a system R₀, and its appearance is as follows:

$$R_0=\left\{\begin{array}{l}x{\wedge }_{0}y\equiv \left(x+y-\sqrt{x^2+y^2}\right),\\ x{\vee }_{0}y\equiv \left(x+y+\sqrt{x^2+y^2}\right),\\ \bar{x}\equiv -x.\end{array}\right.$$

At α = 1, we have a system R₁, and its appearance is as follows:

$$R_1=\left\{\begin{array}{l}x{\wedge }_{1}y\equiv {\displaystyle \frac{1}{2}}\left(x+y-\left|x-y\right|\right),\\ x{\vee }_{1}y\equiv {\displaystyle \frac{1}{2}}\left(x+y+\left|x-y\right|\right),\\ \bar{x}\equiv -x.\end{array}\right.$$

In the last case of R-functions, the conjunctions and disjunctions correspond to:

$$x\wedge y\equiv \mathrm{m}\mathrm{i}\mathrm{n}(x,y), x\vee y\equiv \mathrm{m}\mathrm{i}\mathrm{n}(x,y)$$

Using the R-function, it is possible to construct an implicit form of boundary equations of objects of complex structure based on known equations of simple spheres.

R-functions are used in solving a wide range of mathematical and physics problems, multidimensional digital processing of signals and images, computer graphics, and other fields. Methods for constructing field geometry equations provide a solid technological foundation for automating the process of organizing these equations. In fact, only the process of constructing predicate equations needs to be automated. The transition from these equations to simple elementary equations of field geometry is carried out by replacing the symbols of the logical function with the corresponding symbols of the R-function. However, the symbols of the field are not equal to their corresponding left parts.

So, the input data for the algorithm are following:

• appearance of the standard primitives used: straight line, circle, ellipse, rectangle, triangle, convex polygon, circle, regular polygon, etc.;
• geometric parameters that determine the size and position of standard primitives.

While the dimensions of classic fractals, such as Koch curves, Cayley trees, and Cantor sets, are widely known, we employed the R-function method to derive these dimensions systematically. This method allows us to compute fractal dimensions by integrating geometric modeling with analytical logical functions. Unlike traditional approaches, the R-function framework provides a robust mechanism to model and validate the self-similar properties of fractals iteratively. For instance, the fractal dimension of the Cantor set was calculated based on its recursive construction, with a detailed derivation presented in Equation (12). Similarly, the derivation for the Cayley tree utilized the iterative scaling and branching properties inherent to its structure. These derivations demonstrate the alignment of our methodology with existing formulas while showcasing the adaptability of the R-function approach to various fractal types.

To illustrate the step-by-step process of fractal image generation, we provide the pseudo-code for the recursive fractal modeling (RFM) method. Algorithm 1 iteratively applies transformations to construct complex fractal structures while maintaining symmetry, scaling adaptability, and computational efficiency.

Algorithm 1. Pseudo-code for Fractal Image Generation

Function GenerateFractal(iteration, max_depth, transformation_matrix):

# Base case: Stop recursion if max depth is reached

return

# Apply transformations (scaling, rotation, symmetry preservation)

transformed_points = ApplyTransformation(transformation_matrix)

# Symmetry-preserving transformations

if MaintainSymmetry:

transformed_points = EnforceSymmetry(transformed_points)

# Selective refinement (increase recursion depth for high-detail regions)

if NeedsRefinement(transformed_points):

new_depth = AdjustRecursionDepth(iteration)

else:

new_depth = iteration + 1

# Recursive call to generate the next fractal iteration

for point_set in transformed_points:

GenerateFractal(new_depth, max_depth, transformation_matrix)

# Render the fractal at the current stage

DrawFractal(transformed_points)

# Main function to initialize fractal generation

Function Main():

max_depth = SetMaxRecursionDepth()

transformation_matrix = InitializeTransformations()

GenerateFractal(0, max_depth, transformation_matrix)

For fractal antennas based on a Cayley tree, a fractal antenna represents a series of pieces of conductors of different lengths. At each new iteration, pieces of a certain length are added to the antenna. That is, in each odd iteration, the length remains the same, and in even iterations, the length is halved (see Figure 9). The current distribution in the sixth-order Cayley tree antenna was studied, and new parts of the abbreviation play a role in formalizing the antenna parameters.

Now, based on the RFM method, we will construct the Cayley tree equation.

Step 1.

$$\begin{array}{c}i=1, a_1=l/2, b_1=l/2.\\ f_{oe}(x,y)={\displaystyle \frac{a_{11}^2-(x+a_1{)}^2}{2a_{11}}}\ge 0,(a_{11}—a \mathit{small} \mathit{number}),\\ f_{op}={\displaystyle \frac{a_{11}^2-(a_1-x{)}^2}{2a_{11}}}\ge 0,{\phi }_0(x,y)={\displaystyle \frac{b_1^2-y^2}{2b_1}}\ge 0,\\ f_1=f_{oe}(x,y)\wedge {\phi }_0(x,y)\ge 0,\\ f_2(x,y)=f_{op}(x,y){\wedge }_0{\phi }_0(x,y)\ge 0,\\ {\omega }_{01}(x,y)=f_1(x,y){\vee }_0{f}_2(x,y)\ge 0,\\ f_3\left(x,y\right)={\displaystyle \frac{a_1^2-x^2}{2a_1}}\ge 0,\\ f_4(x,y)={\displaystyle \frac{b_{11}^2-y^2}{2b_{11}}}\ge 0.\end{array}$$

(b₁₁—a sufficiently small number),

$$\begin{array}{c}{\omega }_{02}(x,y)=f_3(x,y){\wedge }_0{f}_4(x,y)\ge 0,\\ f_{1ay}\left(x,y\right)={\displaystyle \frac{b_{11}^2-(y+b_1{)}^2}{2b_{11}}}\ge 0,\\ f_{1by}\left(x,y\right)={\displaystyle \frac{b_{11}^2-(b-y{)}^2}{2b_{11}}}\ge 0, c=b_1/2,\\ {\phi }_{1\mathcal{l}x}\left(x,y\right)={\displaystyle \frac{c^2-(x+a_1{)}^2}{2c}}\ge 0,\\ {\phi }_{1px}={\displaystyle \frac{c^2-(x-a_1{)}^2}{2c}}\ge 0,\\ {\omega }_1(x,y)={\omega }_{01}(x,y){\vee }_0{\omega }_{02}(x,y){\vee }_0{\omega }_{03}(x,y)\ge 0.\end{array}$$

Step 2.

$$\begin{array}{c}i=2, a_1=a_1/2, b_1=b_1/2.\\ {\omega }_2(x,y)={\omega }_1(x-a_1,y-b_1){\vee }_0{\omega }_1(x+a_1,y-b_1){\vee }_0\\ {\vee }_0{\omega }_1(x+a_1,y+b_1){\vee }_0{\omega }_1(x-a_1,y+b_1){\vee }_0{\omega }_1(x,y)\ge 0.\end{array}$$

Now, we construct an iterative process, and as a result, we have the following:

$$\begin{array}{c}i=k, a_1=a_1/2, b_1=b_1/2.\\ {\omega }_k\left(x,y\right)={\omega }_{k-1}\left(x-a_1,y-b_1\right){\vee }_0{\omega }_{k-1}\left(x+a_1,y-b_1\right){\vee }_0{\omega }_{k-1}\left(x+a_1,y+b_1\right){\vee }_0{\omega }_{k-1}\left(x-a_1,y+b_1\right){\vee }_0{\omega }_{k-1}\left(x,y\right)\ge 0,\\ k=3,4,5,\dots \end{array}$$

(2)

Calculation results at different values of k are presented in Figure 9.

For exclusive fractal circles, in the study of fractal circle structures, the properties of symmetry and similarity were considered. In the first iteration, a circle with a radius of 11 mm, a thickness of 0.4 mm along the Ox axis, and a secondary radius of 0.2 mm was considered as the main element of the fractal antenna structure in A1. The algorithm for constructing the structure of the fractal abbreviation represented in Figure 10 is expressed as follows.

In iteration 0, seven circles that are three times smaller than the given circle are placed in the main circle. Other elements (width and thickness of the circle) stay unchanged. The center of the six small circles is placed at a distance R*2/3 from the tip of the hexagon. The center of the seventh circle coincides with the center of the main antenna. We call this construction the first step of the iterative algorithm and denote it by the abbreviator A1.

The algorithm used for model A1 is used to construct exclusive circles A2 in the second iteration (Figure 10).

For each circle, six smaller circles are placed, each with a radius half of the previous one, with centers located at R*2/3 from the center of the initial circle, positioned at the vertices of a hexagon. A seventh circle is centered within the main circle. It forms the fractal antenna model shown in Figure 11. The coaxial lines have a diameter of 0.5 mm, an antenna thickness of 0.4 mm, and a ring width of 0.2 mm. The outer circle radiuses are R = 11 mm, R1 = R/3, and R2 = R/9.

We will construct the equation for the exclusive antenna A1:

Step 0.

$${\omega }_{00}={\displaystyle \frac{R_0^2-x^2-y^2}{2R}}.$$

(3)

Step 1.

$$\begin{array}{c}{r}_1={\displaystyle \frac{1}{3}}{R}_0, a_1={\displaystyle \frac{2}{3}}{R}_0, dx={\displaystyle \frac{\sqrt{3}}{3}}{R}_0, dy={\displaystyle \frac{1}{3}}{R}_0,\\ {\omega }_{10}={\displaystyle \frac{r_1^2-x^2-y^2}{2r_1}}, {\omega }_{11}={\displaystyle \frac{r_1^2-x^2-(y-a_1{)}^2}{2r_1}},\\ {\omega }_{12}={\displaystyle \frac{r_1^2-x^2-(y+a_1{)}^2}{2r_1}}, {\omega }_{14}={\displaystyle \frac{r_1^2-(x+dx{)}^2-(y+dy{)}^2}{2r_1}},\\ {\omega }_{15}={\displaystyle \frac{r_1^2-(x-dx{)}^2-(y+dy{)}^2}{2r_1}}, {\omega }_{16}={\displaystyle \frac{r_1^2-(x-dx{)}^2-(y-dy{)}^2}{2r_1}},\\ \omega =\left({\omega }_{00}{\vee }_0\right({\omega }_{10}{\vee }_0{\omega }_{11}{\vee }_0{\omega }_{12}{\vee }_0{\omega }_{13}{\vee }_0{\omega }_{14}{\vee }_0{\omega }_{15}{\vee }_0{\omega }_{16}\left)\right).\end{array}$$

(4)

This equation gives the equation of the exclusive antenna A1.

dr is a small number—the thickness of the circle.

$${\omega }_0={\displaystyle \frac{R_0^2-x^2-y^2}{2R}}{\wedge }_0{\displaystyle \frac{x^2+y^2-(R-dr{)}^2}{2R}}\ge 0.$$

(5)

Step 1:

$$\begin{array}{c}{r}_1={\displaystyle \frac{1}{3}}{R}_0,a_1={\displaystyle \frac{2}{3}}{R}_0,dx={\displaystyle \frac{\sqrt{3}}{3}}{R}_0,dy={\displaystyle \frac{1}{3}}{R}_0.\\ {\omega }_{10}={\omega }_0(r_1,x,y),\\ {\omega }_{11}={\omega }_0(r_1,x,y-a_1),\\ {\omega }_{12}={\omega }_0(r_1,x,y+a_1),\\ {\omega }_{13}={\omega }_0\left(r_1,x+dx,y-dy\right),\\ {\omega }_{14}={\omega }_0(r_1,x+dx,y+dy),\\ {\omega }_{15}={\omega }_0(r_1,x-dx,y+dy),\\ {\omega }_{16}={\omega }_0(r_1,x-dx,y-dy),\\ {\omega }_1=\left({\omega }_{00}{\vee }_0\right({\omega }_{10}{\vee }_0{\omega }_{11}{\vee }_0{\omega }_{12}{\vee }_0{\omega }_{13}{\vee }_0{\omega }_{14}{\vee }_0{\omega }_{15}{\vee }_0{\omega }_{16}\left)\right).\end{array}$$

We determine the i-step from all of the previous ones [23]:

$$\begin{array}{c}{r}_i={\displaystyle \frac{1}{3}}{r}_{i-1},a_i={\displaystyle \frac{2}{3}}{r}_{i-1},dx={\displaystyle \frac{\sqrt{3}}{3}}{r}_{i-1},dy={\displaystyle \frac{1}{3}}{r}_{i-1}.\\ {\omega }_{i0}={\omega }_{i-1}(r_i,x,y),\\ {\omega }_{i1}={\omega }_{i-1}\left(r_i,x,y-a_i\right),\\ {\omega }_{i2}={\omega }_{i-1}(r_i,x,y+a_i),\\ {\omega }_{i3}={\omega }_{i-1}(r_i,x+dx,y-dy),\\ {\omega }_{i4}={\omega }_{i-1}(r_i,x+dx,y+dy),\\ {\omega }_{i5}={\omega }_{i-1}(r_i,x-dx,y+dy),\\ {\omega }_{i6}={\omega }_{i-1}(r_i,x-dx,y-dy),\\ \omega =\left({\omega }_{00}{\vee }_0\right({\omega }_{i0}{\vee }_0{\omega }_{i1}{\vee }_0{\omega }_{i2}{\vee }_0{\omega }_{i3}{\vee }_0{\omega }_{i4}{\vee }_0{\omega }_{i5}{\vee }_0{\omega }_{i6}\left)\right).\end{array}$$

(6)

It can be noted that we obtain the model of antenna A₂ at i = 2, i = 0, 1, 2, 3, 4. The results of the calculations are presented in Figure 11.

For the Serpin curve, first, we construct the equation of the base:

$$\begin{array}{c}{x}_1=x\mathit{cos}({\alpha }_1)+y\mathit{sin}({\alpha }_1), y_1=-x\mathit{sin}({\alpha }_1)+y\mathit{cos}({\alpha }_1),\\ x_2=x\mathit{cos}({\alpha }_2)+y\mathit{sin}({\alpha }_2), y_2=-x\mathit{sin}({\alpha }_2)+y\mathit{cos}({\alpha }_2),\\ f_1\left(x,y\right)=\left(a^2-{x_1}^2\right){\wedge }_0\left(a^2-{y_1}^2\right)\ge 0,\\ f_2\left(x,y\right)=\left(a^2-{x_2}^2\right){\wedge }_0\left(a^2-{y_2}^2\right)\ge 0,\\ f_3\left(x,y\right)=f_2\left(-x,y\right),\\ {\omega }_1(x,y)=f_1(x,y){\vee }_0{f}_2(x,y){\vee }_0{f}_3(x,y).\end{array}$$

(7)

Here, we used the formulas for turning the axes. We construct an iterative process, and as a result, we have the following:

$$\begin{array}{c}{\omega }_n\left(x,y\right)={\omega }_{n-1}\left(x,y\right){\vee }_0{\omega }_{n-1}\left(x-2a,y-2a\right){\vee }_0{\omega }_{n-1}\left(x+2a,y-2a\right){\vee }_0{\omega }_{n-1}\left(x+2a,y+2a\right){\vee }_0{\omega }_{n-1}\left(x-2a,y+2a\right),\\ n=2, 3, 4, 5,\dots \end{array}$$

(8)

We used $a_1={\displaystyle \frac{3}{8}}a,b_1={\displaystyle \frac{7}{4}}a$ in the calculation. The results of calculations in α₁ = 0 and α₂ = π/4 and the different values are presented in Figure 12.

For the Koch curve, the Koch curve is constructed by the following procedure. The center piece from the initial span is removed and replaced with an equilateral triangle. Then, the same procedure is applied to all the resulting triangles.

Let the half-plane y ≤ 0 be the initial domain. Then, its boundary equation is ω₀ = −y = 0. The algorithm for finding the equation is (k ≥ 1); k-ordered prefractals based on the RFM method consist of two stages:

1. R-disjunction of the function ω_(k−1) with the reflection of itself with respect to the straight line;

$$\begin{array}{c}y=-x\sqrt{3}+3^{k-1}\sqrt{3};\\ {\varpi }_k(x,y)={\omega }_{k-1}(x,y){\vee }_0{\omega }_{k-1}\left[\left(-x-y\sqrt{3}+3^k\right)/2,\left(-x\sqrt{3}+y+3^{k-1}\sqrt{3}\right)/2\right].\end{array}$$

(9)

2. R-conjugation of the function ϖ_k with its reflection with respect to the line.

$$\begin{array}{c}x=3^k/2:\\ {\omega }_k(x,y)={\varpi }_k(x,y){\wedge }_0{\varpi }_k(3^k-x,y).\end{array}$$

(10)

A Koch curve constructed on the sides of equilateral triangles gives a geometric object called a Koch snowflake. The boundary equations of the Koch snowflake are obtained by applying the three operations’ R-conjunction, which are 60° degrees from each other:

$$\begin{array}{ll}{W}_k(x,y)=& {\omega }_k(y+3^k/2,x\\ & -3^{k-1}\sqrt{3}\\ & /2\left){\wedge }_0{\wedge }_0\right[{\omega }_k\left.y-x\sqrt{3}+3^k\right)/2,\left(-x-y\sqrt{3}-3^{k-1}\sqrt{3}\right)\\ & /2]{\wedge }_0\left.{\omega }_k\left[\left(-y-x\sqrt{3}+3^k\right)/2,\left(-x+y\sqrt{3}-3^{k-1}\sqrt{3}\right)/2\right]\right].\end{array}$$

(11)

For the Cantor set, the Cantor set is a fundamental example in fractal geometry with significant properties. It is constructed as follows.

For the starting set, begin with the interval [0, 1].

The first step is to remove the middle third of this interval, which is the open interval (1/3, 2/3). The remaining set is two intervals: [0, 1/3] and [2/3, 1].

For the subsequent steps, in each following step, remove the middle third of each remaining interval from the previous step, excluding the endpoints. By iteratively removing the middle third from each interval, a Cantor set is formed, which has a dimension of approximately d ≈ 0.9542.

A Cantor set of dimension d = 1 can be constructed by moving from a straight line to a plane. We assume that the initial set is a unit square, and the vertices are (0, 0), (1, 0), (1, 1), and (0, 1). At each step, we create squares that are four times smaller than the previous square. The limiting set of this construction is a self-similar set with N = 4 and a similarity coefficient r = 1/4. Its size is equal to $d=\mathit{log}(4)/\mathit{log}(4)=1.$

As can be seen from the constructions, the resulting set is a Cantor set, which is compact, perfect, and completely continuous. We construct a quadratic equation for the equation of this fractal using V. L. Rvachev’s RFM method.

Based on the condition of the Cantor set of size d = 1, we build an iterative process. As a result, we attain the following, where n = 1, 2, 3, …:

$$\begin{array}{ll}{\omega }_n(a,x,y)=& {\omega }_{n-1}\left({\displaystyle \frac{a}{4}},x-{\displaystyle \frac{3a}{4}},y-{\displaystyle \frac{a}{4}}\right){\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{a}{4}},x-{\displaystyle \frac{a}{4}},y-{\displaystyle \frac{3a}{4}}\right){\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{a}{4}},x+{\displaystyle \frac{3a}{4}},y+{\displaystyle \frac{a}{4}}\right){\vee }_0{\omega }_{n-1}({\displaystyle \frac{a}{4}},x\\ & +{\displaystyle \frac{a}{4}},y+{\displaystyle \frac{3a}{4}}){\vee }_0\left(x\left(a^2-x^2\right)\right.=\left.0{\wedge }_0\left(a^2-y^2\right)\ge 0\right){\vee }_0\left(y\left(a^2-y^2\right)\right.\\ & =0{\wedge }_0\left.\left(a^2-x^2\right)\ge 0\right),\end{array}$$

(12)

The results obtained when d = 1 and the different values of n are presented in Figure 13.

For the fractals of circles, currently, there are several methods of constructing the fractal equation: the L-systems method, the iterative function systems method, etc. Different from them, the design environment of the R-function algebraic logic method makes it possible to construct the fractal equation. A visual representation of fractals can then be constructed from these equations. Thus, the construction of equations of fractals consisting of circles is based on the RFM method considered below. The equation of the outer circle is defined as:

$${\omega }_{00}={\omega }_{00}(R,x,y)=(R^2-x^2-y^2\ge 0),$$

The equation of a circle connected to a circle takes the following form:

$${\omega }_0={\omega }_{00}{\wedge }_0(x^2+y^2-(R-a{)}^2\ge 0),$$

where a is the thickness of the circle (thickness of the circle is 2a), R is the radius of the outer circle, α = 2π/k; and k is the number of inner circles after each iteration k = 2, 3, 4, …

Using iteration here, we attain the following:

$$\begin{array}{c}{\omega }_n\left(R,x,y\right)={\omega }_0\left(R,x,y\right){\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{3}},x,y\right){\vee }_0\\ {\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{3}},x-{\displaystyle \frac{2R}{3}}\cos \left(0\right),y-{\displaystyle \frac{2R}{3}}\sin \left(0\right)\right){\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{3}},x-{\displaystyle \frac{2R}{3}}\cos \left(\alpha \right),y-{\displaystyle \frac{2R}{3}}\sin \left(\alpha \right)\right){\vee }_0{\omega }_{n-1}({\displaystyle \frac{R}{3}},x-\\ {\displaystyle \frac{2R}{3}}\cos \left(2\alpha \right),y-{\displaystyle \frac{2R}{3}}\sin \left(2\alpha \right)){\vee }_0..{\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{3}},x-{\displaystyle \frac{2R}{3}}\cos \left(\left(k-1\right)\alpha \right),y-{\displaystyle \frac{2R}{3}}\sin \left(\left(k-1\right)\alpha \right)\right)\ge 0; n=1 ,2 ,3 \dots \end{array}$$

(13)

It can be distinguished that, if in k < 6, the inner circles do not touch each other, in k = 6 the inner circles do, and in k > 6, the inner circles intersect, as presented in Figure 14.

For intersecting circular fractals, let us look at the case where there are two circles inside of a big circle. In turn, two more circles are formed in the inner circles, and so on. We construct the same fractal equation. In this case, the equation of the fractal in step 1 takes the following form:

$$\omega (R,x,y)=(R^2-x^2-y^2\ge 0){\wedge }_0(x^2+y^2-(R-a{)}^2\ge 0).$$

(14)

where α is the thickness of the circle (the thickness of the circle is equal to 2α), and R is the radius of the outer circle. After applying the iteration procedure, we generate the following:

$${\omega }_n\left(R,x,y\right)={\omega }_0\left(R,x,y\right){\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{2}},x,y-{\displaystyle \frac{R}{2}}\right){\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{2}},x,y+{\displaystyle \frac{R}{2}}\right)\ge 0; n=1, 2, 3, \dots$$

(15)

The calculation results for different values of n are presented in Figure 15.

Now, we consider the case where the inner circles intersect and decrease. For this purpose, we introduce the decreasing coefficient l. We determine the equation of intersecting circles, as in the first problem.

$${\omega }_0(R,x,y)=(R^2-x^2-y^2\ge 0){\wedge }_0(x^2+y^2-(R-a{)}^2\ge 0)$$

(16)

where a is the thickness of the circle (the thickness of the circle is equal to 2a).

$$a={\displaystyle \frac{2\pi }{k}}$$

K is the number of inner loops after each iteration k = 2, 3, 4, …, and l is the reduction coefficient of inner circles after each iteration, l = 2, 3, 4, …; R is the radius of the outer circle. Using the iteration procedure, we attain the following [1]:

$$\begin{array}{ll}{\omega }_n\left(R,x,y\right)=& {\omega }_0\left(R,x,y\right)\\ & {\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{l}},x,y\right){\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{l}},x-{\displaystyle \frac{\left(l-1\right)R}{l}}\cos \left(0\right),y-{\displaystyle \frac{\left(l-1\right)R}{l}}\sin \left(0\right)\right){\vee }_0{\omega }_{n-1}({\displaystyle \frac{R}{l}},x\\ & -{\displaystyle \frac{\left(l-1\right)R}{l}}\cos \left(\alpha \right),y-{\displaystyle \frac{\left(l-1\right)R}{l}}\sin \left(\alpha \right)\left){\vee }_0{\omega }_{n-1}\right({\displaystyle \frac{R}{l}},x-{\displaystyle \frac{\left(l-1\right)R}{l}}\cos \left(2\alpha \right),y\\ & -{\displaystyle \frac{\left(l-1\right)R}{l}}\sin \left(2\alpha \right)){\vee }_0{\omega }_{n-1}\left({\displaystyle \frac{R}{l}},x-{\displaystyle \frac{\left(l-1\right)R}{l}}\cos \left(\left(k-1\right)\alpha \right),y-{\displaystyle \frac{\left(l-1\right)R}{l}}\sin \left(\left(k-1\right)\alpha \right)\right)\\ & \ge 0; n=1, 2, 3, \dots \end{array}$$

(17)

The results for different values of n, k, l and the iterative fractals are obtained at k = 8, l = 2 and $n=\overline{1,2,3}$. It is represented in Figure 16.

For fractals in a tree view, known geometric fractals are formalized starting from an initiator shape using a basic image. Determinized fractals are expressed in a recursive process. In deterministic fractals, self-similarity appears in all orders. Such fractals are iterated 4–6 times to obtain clear images.

In this section, we will construct a tree-like fractal equation based on the R-functions method. Let us look at constructing a tree equation from the circles. Let the ends of the interval be the points (x₁, y₁) and (x₂, y₂). We construct the equation of a straight line freely passing through the given points (x₁, y₁) and (x₂, y₂) [1].

$$\begin{array}{l}f\left(x_1,y_1,x_2,y_2,x,y\right)\\ & ={\left(\left({\displaystyle \frac{1}{2}}\left(\right(x_2-x_1\right)\mathit{cos}\left(\mathit{arctan}\left({\displaystyle \frac{y_2-y_1}{x_2-x_1}}\right)\right)+\left(y_2-y_1\right)\mathit{sin}\left(\mathit{arctan}\left({\displaystyle \frac{y_2-y_1}{x_2-x_1}}\right)\right)\right)}^2\\ & -\left(\left(x-x_1\right)\mathit{cos}\left(\mathit{arctan}\left({\displaystyle \frac{y_2-y_1}{x_2-x_1}}\right)\right)+\left(y-y_1\right)\mathit{sin}\left(\mathit{arctan}\left({\displaystyle \frac{y_2-y_1}{x_2-x_1}}\right)\right)\right.\\ & -\left({\displaystyle \frac{1}{2}}\left(x_2-x_1\right)\cos \left(\arctan \left({\displaystyle \frac{y_2-y_1}{x_2-x_1}}\right)+\left(y_2-y_1\right)\sin {\left(\left(\arctan \left({\displaystyle \frac{y_2-y_1}{x_2-x_1}}\right)\right)\right)}^2\ge 0\right)\right.(a^2\\ & -\left(-\left(x-x_1\right)\mathit{sin}\left(\mathit{arctan}\left({\displaystyle \frac{y_2-y_1}{x_2-x_1}}\right)\right)\right)+\left(y-y_1\right)\mathit{cos}\left.{\left.\left(\mathit{arctan}\left({\displaystyle \frac{y_2-y_1}{x_2-x_1}}\right)\right)\right)}^2\ge 0\right)\end{array}$$

(18)

where a is the height of the gap (the height of the gap is equal to 2a). If k is even, then ${\phi }_0=0$. Otherwise, ${\phi }_0={\displaystyle \frac{\alpha }{2}}$.

At n = 1 we have the following equation: α = 2π/k

$$\begin{array}{c}{\omega }_1(x,y)=f(0,0,R\mathit{cos}({\phi }_0+0),R\mathit{sin}({\phi }_0+0),x,y)f(0,0,R\mathit{cos}({\phi }_0+\alpha ),R\mathit{sin}({\phi }_0+\alpha ),x,y){\vee }_0\\ {\vee }_{0}f(0,0,R\mathit{cos}({\phi }_0+2\alpha ),R\mathit{sin}({\phi }_0+2\alpha ),x,y){\vee }_0\dots {\vee }_{0}f(0,0,R\mathit{cos}({\phi }_0+(k-1\left)\alpha \right),R\mathit{sin}({\phi }_0+(k-1)\alpha ),x,y)\end{array}$$

(19)

At n = 2, 3, 4…

$$\alpha ={\displaystyle \frac{2\pi }{k^{n-1}}};{ k}_1=-\left[{\displaystyle \frac{k}{2}}\right]; R_{n-1}=2R\left(1-{\displaystyle \frac{1}{2^{n-1}}}\right); R_n=2R(1-{\displaystyle \frac{1}{2^n}});$$

R_n is the radius of the circular boundaries in the n-th iteration (R1 = R). If k is even, then k₂ = [k/2]. Otherwise, k₂ = [k/2] − 1. [x] is an integer part of the number x. Using the iteration procedure, we attain the following:

$$\begin{array}{ll}{\omega }_{nx1}(x,y)=& f(R_{n-1}\mathit{cos}({\phi }_0+\alpha ),R_{n-1}\mathit{sin}({\phi }_0\\ & +\alpha ),R_n\mathit{cos}\left({\phi }_0+\alpha +{\displaystyle \frac{\left({\phi }_0+k_1\alpha \right)}{k}}\right),R_n\mathit{sin}\left({\phi }_0+\alpha +{\displaystyle \frac{\left({\phi }_0+k_1\alpha \right)}{k}}\right),x,y)\\ & {\vee }_0 f\left(R_{n-1}\mathit{cos}({\phi }_0+\alpha \right),R_{n-1}\mathit{sin}({\phi }_0\\ & +\alpha ),R_n\cos \left({\phi }_0+\alpha +{\displaystyle \frac{\left({\phi }_0+\left(k_1+1\right)\alpha \right)}{k}}\right),R_n\sin ({\phi }_0+\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+\left(k_1+1\right)\alpha \right)}{k}}),x,y){\vee }_{0}f\left(R_{n-1}\cos \left({\phi }_0+\alpha \right),R_{n-1}\sin \left({\phi }_0+\alpha \right),\right.R_n\mathit{cos}({\phi }_0+\alpha \\ & +{\displaystyle \frac{({\phi }_0+(k_1+2\left)\alpha \right)}{k}}),R_n\mathit{sin}({\phi }_0+\alpha \\ & +{\displaystyle \frac{({\phi }_0+(k_1+2\left)\alpha \right)}{k}}),x,y){\vee }_0\dots {\vee }_{0}f(R_{n-1}\cos \left({\phi }_0+\alpha \right),R_{n-1}\sin \left({\phi }_0+\alpha \right),R_n\cos ({\phi }_0+\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+k_2\alpha \right)}{k}}),R_n\sin ({\phi }_0+\alpha +{\displaystyle \frac{({\phi }_0+k_2\alpha )}{k}}),x,y)\end{array}$$

$$\begin{array}{ll}{\omega }_{nx2}(x,y)=& f(R_{n-1}\mathit{cos}({\phi }_0+2\alpha ),R_{n-1}\mathit{sin}({\phi }_0+2\alpha ),R_n\mathit{cos}({\phi }_0+2\alpha +{\displaystyle \frac{\left({\phi }_0+k_1\alpha \right)}{k}}),R_n\mathit{sin}({\phi }_0+2\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+k_1\alpha \right)}{k}}),x,y){\vee }_{0}f(R_{n-1}\mathit{cos}({\phi }_0+2\alpha ),R_{n-1}\mathit{sin}({\phi }_0+2\alpha ),R_n\mathit{cos}({\phi }_0+2\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+\left(k_1+1\right)\alpha \right)}{k}}),R_n\mathit{sin}({\phi }_0+2\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+\left(k_1+1\right)\alpha \right)}{k}}),x,y){\vee }_{0}f (R_{n-1}\cos \left({\phi }_0+2\alpha \right),R_{n-1}\sin \left({\phi }_0+2\alpha \right),R_n\cos ({\phi }_0+2\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+\left(k_1+2\right)\alpha \right)}{k}}),R_n\sin \left({\phi }_0+2\alpha +{\displaystyle \frac{\left({\phi }_0+\left(k_1+2\right)\alpha \right)}{k}}\right),x,y){\vee }_0\dots {\vee }_0 f(R_{n-1}\mathit{cos}({\phi }_0\\ & +2\alpha ),R_{n-1}\mathit{sin}({\phi }_0+2\alpha ),R_n\mathit{cos}({\phi }_0+2\alpha +{\displaystyle \frac{\left({\phi }_0+k_2\alpha \right)}{k}}),R_n\mathit{sin}({\phi }_0+2\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+k_2\alpha \right)}{k}}),x,y)\end{array}$$

For 1 ≤ i ≤ kⁿ⁻¹, we have:

$$\begin{array}{c}\begin{array}{ll}{\omega }_{nxi}(x,y)=& f(R_{n-1}\cos \left({\phi }_0+i\alpha \right),R_{n-1}\sin \left({\phi }_0+i\alpha \right),R_n\cos ({\phi }_0+i\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+k_1\alpha \right)}{k}}),R_n\sin \left({\phi }_0+i\alpha +{\displaystyle \frac{\left({\phi }_0+k_1\alpha \right)}{k}}\right),x,y){\vee }_0 f(R_{n-1}\cos \left({\phi }_0+i\alpha \right),R_{n-1}\sin ({\phi }_0\\ & +i\alpha )R_n\cos \left({\phi }_0+i\alpha +{\displaystyle \frac{\left({\phi }_0+\left(k_1+1\right)\alpha \right)}{k}}\right),R_n\sin ({\phi }_0+i\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+\left(k_1+1\right)\alpha \right)}{k}}),x,y){\vee }_{0}f(R_{n-1}\cos \left({\phi }_0+i\alpha \right),R_{n-1}\sin \left({\phi }_0+i\alpha \right)R_n\cos ({\phi }_0+i\alpha \\ & +{\displaystyle \frac{\left({\phi }_0+\left(k_1+2\right)\alpha \right)}{k}}),R_n\sin \left({\phi }_0+i\alpha +{\displaystyle \frac{\left({\phi }_0+\left(k_1+2\right)\alpha \right)}{k}}\right),x,y){\vee }_0\dots {\vee }_{0}f(R_{n-1}\cos ({\phi }_0\\ & +i\alpha ),R_{n-1}\sin \left({\phi }_0+i\alpha \right)R_n\cos \left({\phi }_0+i\alpha +{\displaystyle \frac{\left({\phi }_0+k_2\alpha \right)}{k}}\right),R_n\sin \left({\phi }_0+i\alpha +{\displaystyle \frac{\left({\phi }_0+k_2\alpha \right)}{k}}\right),x,y)\end{array}\\ {\omega }_n\left(x,y\right)={\omega }_{n-1}\left(x,y\right){\vee }_0{\omega }_{nx1}\left(x,y\right){\vee }_0{\omega }_{nx2}\left(x,y\right){\vee }_0\dots {\vee }_0{\omega }_{nxi}\left(x,y\right){\vee }_0{\vee }_0\dots {\vee }_0{\omega }_{nx{k}^{n-1}}\left(x,y\right).\end{array}$$

(20)

In the previous formulas, the values were k = 2, 3, 4, 5, ….

For all lines, an outer circle with a radius Rn can be drawn (n-order iteration). The calculation results for the different values of n and k are presented in Figure 17.

For a Pythagorean tree, Pythagoras, proving his theorem, constructed a figure in which squares are placed on the sides of right triangles. If this process is continued, a Pythagorean tree is formed. We construct the equation of the tree using quadratic equations, i.e.,

$$\begin{array}{ll}{\omega }_0(a,x,y)=& \left(\right(a^2-x^2\ge 0\left){\wedge }_0\right((b^2-(y-a{)}^2\ge 0\left){\vee }_0\right(b^2-(y+a{)}^2\ge 0)\left)\right){\vee }_0\left(\right(a^2-y^2\\ & \ge 0\left){\wedge }_0\right((b^2-(x-a{)}^2\ge 0\left){\vee }_0\right(b^2-(x+a{)}^2\ge 0)\left)\right)\ge 0\end{array}$$

where ${\omega }_0(a,x,y)$ is a square with side 2a and thickness 2b. Using the recursion procedure, we create the following:

$$\begin{array}{ll}{\omega }_n(a,x,y)=& {\omega }_0(a,x,y){\vee }_0{\omega }_{n-1}(a\mathit{cos}(\alpha ),(x+a-a\sqrt{2}\mathit{cos}(\alpha )\mathit{sin}({\displaystyle \frac{\pi }{4}}-\alpha ))\mathit{cos}(\alpha )+(y-a\\ & -a\sqrt{2}\mathit{cos}(\alpha )\mathit{cos}({\displaystyle \frac{\pi }{4}}-\alpha ))\mathit{sin}(\alpha ),-(x+a-a\surd 2\mathit{cos}(\alpha )\mathit{sin}({\displaystyle \frac{\pi }{4}}-\alpha ))\mathit{sin}(\alpha )+(y-a\\ & -a\surd 2\mathit{cos}(\alpha )\mathit{cos}({\displaystyle \frac{\pi }{4}}-\alpha ))\mathit{cos}(\alpha )){\vee }_0{\omega }_{n-1}(a\mathit{sin}(\alpha ),-(x-a-a\sqrt{2}\mathit{sin}(\alpha )\mathit{sin}({\displaystyle \frac{\pi }{4}}\\ & -\alpha ))\mathit{sin}(\alpha )+(y-a-a\sqrt{2}\mathit{sin}(\alpha )\mathit{cos}({\displaystyle \frac{\pi }{4}}-\alpha ))\mathit{cos}(\alpha ),(x-a-a\sqrt{2}\mathit{sin}(\alpha )\mathit{sin}({\displaystyle \frac{\pi }{4}}\\ & -\alpha ))\mathit{cos}(\alpha )+(y-a-a\sqrt{2}\mathit{sin}(\alpha )\mathit{cos}({\displaystyle \frac{\pi }{4}}-\alpha ))\mathit{sin}(\alpha ))\end{array}$$

(21)

where α is the turning angle of the tree branch when turning to the left takes the value into the interval 0 < α < π/2, and the turning angle when turning to the right is equal to π/2 − α. The results of calculations at different values of n and α are presented in Figure 18.

Tree-shaped fractals are considered the simplest fractals. The equation of various tree-shaped fractals can be constructed using the equation of a straight line and the design environment of the RFM method, namely the R₀: R-conjunction, R-disjunction, and R-negation. Based on these equations, by giving the number of iterations and the twist angle α, it is possible to create various prefractals used in computer landscapes, various illustrations, the textile industry, etc.

In fractal construction methods, equations for classical and modern fractals are constructed, results are obtained, and pictures are presented.

The equation of circles and the algebra logical method can construct a fractal equation from the intersection of circles and the union of circles using the R-function design tool. These fractals are very beautiful and can be used in light industry, telecommunications, drawing patterns on ceramics and porcelain, etc. Multiple theoretical studies on logic algebra, the RFM method, and fractal arithmetic methods serve to develop an algorithmic environment for modernizing the color design of carpets and rugs.

3. Visualization Algorithm of Patterns with Complex Fractal Structure

The algorithm for the visualization of dragon and tree-like fractals using the RFM and geometric substitutions is implemented as follows (see Figure 19 and Figure 20):

The rules for constructing a dragon fractal using the L-systems method are as follows:

axiom = X,
newx = X+Y+,
newy = −X−Y.

Here are a few steps to building a dragon using these generative rules:

Step 1: FX+YF+

Step 2: FX+YF++-FX-YF+

Step 3: FX+YF++-FX-YF++-FX+YF+ − −FX-YF+

Step 4: FX+YF++-FX-YF++-FX+YF+ − −FX-YF++ -FX+YF++-FX-YF+ − − FX+YF+ − − FX-YF+

Let us look at the creation of complex fractals using such types of geometric transformations as rotation, translation, and reflection.

For Step 1, twist the result obtained by the L-systems method at an angle α, and this twisting can be continued until the desired result is obtained.

$$\begin{array}{c}{x}^{\prime }=x\mathit{cos}\alpha -y\mathit{sin}\alpha ,\\ y^{\prime }=x\mathit{sin}\alpha +y\mathit{cos}\alpha .\end{array}$$

In this case, a counterclockwise rotation along the starting point of the coordinate system is performed. If the number of steps in this turn is n times, a = 360/n is equal.

For Step 2, geometric transformations are performed using the translation or reflection types of geometric transformations of the result. Reflection in the plane is performed relative to the axes of the coordinate system. The reflection relative to the abscissa axis is expressed as:

$$\begin{array}{c}{x}^{\prime }=x\\ y^{\prime }=-y\end{array}$$

The reflection relative to the ordinate axis is expressed as follows:

$$\begin{array}{c}{x}^{\prime }=-x\\ y^{\prime }=y\end{array}$$

For Step 3, Turning (moving) is performed based on the given angle.

$$\left\{\begin{array}{c}{x}^{\prime }=x+rcos{\displaystyle \frac{\pi d}{180}}\\ y^{\prime }=x+rsin{\displaystyle \frac{\pi d}{180}}\end{array}\right. \mathrm{here} d={\displaystyle \frac{360°}{n}}$$

The process can be continued until the desired result is achieved or the result obtained is satisfactory.

As a result, fractals are formed as follows by performing geometric substitutions using the L-systems method (see Figure 21 and Figure 22):

For the algorithm for constructing fractals from pentagons, the construction of fractals of this type is carried out in the same way as in the above algorithms. First, a pentagon whose side is equal to “a” is drawn, and its center is determined (its center is the point of intersection of the heights passed from the ends). The coordinate of this point is determined (see Figure 23a).

In the next step, the edges of the resulting pentagons are reduced by 2/5 times and placed at the ends of the first pentagon (see Figure 23b).

$$\begin{array}{c}A\left(x,y\right)=\left(x-a/2, y-a\ast 2/3\right);\\ B\left(x,y\right)=\left(x-a\ast 3/5, y-a\ast 2/3\right);\\ C\left(x,y\right)=\left(x+a, y+a/2\right);\\ D\left(x,y\right)=\left(x-a\ast 8/9, y+a/2\right);\\ E\left(x,y\right)=\left(x+a/17, y+a\ast 10/9\right).\end{array}$$

Repeating this process n times (see Figure 24), the formula for calculating the number of pentagons can be written as follows:

$$1+5+25+125+\cdots +5^{n-1}=\sum _{i=1,2\dots }^n{5}^{i-1}$$

The fractal visualization software developed in this research extends beyond typical fractal drawing tools by enabling user-driven customization and the generation of high-resolution patterns. Unlike general-purpose fractal visualizers, our tool integrates tailored functionalities specific to iterated-function systems (IFS), L-systems, and R-function-based fractal models, which allows users to produce a wide range of culturally inspired and application-specific patterns.

The fractal dimension is a fundamental measure of the complexity and self-similarity of fractal structures. Unlike Euclidean dimensions, which are integer values, fractal dimensions provide a quantitative measure of how a fractal pattern fills space across different scales. In this study, we calculated the fractal dimensions of the generated fractals using the box-counting method:

$$D_f=\underset{ϵ\to 0}{\mathit{lim}}{\displaystyle \frac{\mathit{log}N\left(ϵ\right)}{\mathit{log}(1/ϵ)}}$$

where D_f is the fractal dimension, N(ϵ) is the number of boxes of size ϵ required to cover the fractal, and ϵ is the scale of measurement.

The fractal dimensions for selected fractal patterns generated by the RFM method are summarized in

Table 1. Fractal dimensions for selected fractal patterns.

Polygon
	M(500;400), R = 120, L = 1.5, α = 0, n = 10, S = 1	M(500;400), R = 120, L = 1.5, α = 0, n = 10, S = 2	M(500;400), R = 120, L = 1.5, α = 0, n = 16, S = 2
	D1 = 1.11	D2 = 1.59	D3 = 1.79
Circular
	M(500:400), R = 350, α = 45, n = 4, l = 45, k = 2, S = 3	M(500:400), R = 350, α = 0, n = 4, l = 45, k = 2, S = 5	M(500:400), R = 350, α = 15, n = 4, l = 0, k = 2.7, S = 9
	D1 = 1.75	D2 = 1.86	D3 = 1.90
Koch
	Axiom: $F++++F$ General rule: $F\to F+F+F++++F+F+F$ Number of iterations: 4, Scale: 9, Turning angle: 45° Color: purple	Axiom: $F-F-F-F-F$ General rule: $F\to F-F-F++F+F-F$ Number of iterations: 4, Scale: 6, Turning angle: 72° Color: red	Axiom: $F++F++F++F++F$ General rule: $F\to F++F++F+++++F-F++F$ Number of iterations: 6, Scale: 3, Turning angle: 36° Color: #1c0397
	D = 1.53	D = 1.87	D = 1.96
Cayley tree
	n = 2	n = 4	n = 5
	D = 1.67	D = 1.78	D = 1.86
Serpin
	n = 2 (α1 = 0, α2 = π/4)	n = 3 (α1 = 0, α2 = π/4)	n = 4 (α1 = π/4, α2 = π/4)
	D = 1.80	D = 1.94	D = 1.91
Tree-shaped
	Reflection = 1, Reflection of element = 11, Movement = 50, Scale = 6	Reflection = 5, Reflection of element = 6, Movement = 71, Scale = 6	Reflection = 7, Reflection of element = 7, Movement = 70, Scale = 14
	D = 1.89	D = 1.92	D = 1.86

and Appendix A, Figure A6.

The key functionalities of the software are as follows.

For automated scaling and recursion adjustment, users can control recursion depth, scaling factors, and symmetry options, enabling the creation of highly detailed fractal structures that maintain consistency across different resolutions.

For cultural and practical design customizations, the software supports the input of parameters that are reflective of culturally significant designs, making it particularly valuable for traditional pattern construction in textile design and architectural decor.

For industrial output compatibility, by generating high-resolution fractal images, the software caters to industrial applications, such as printing on textiles, ceramics, and architectural materials. The output can be adjusted to meet specific size and resolution requirements, which is often a limitation in general fractal visualization tools.

This tool’s unique feature set provides a versatile platform for both visualization and practical implementation, thus enhancing its functionality beyond that of existing fractal tools and addressing the needs of specific design-oriented applications.

4. Comparison of Performance, Usability, and Efficiency

In this section, we highlight the distinctive efficiency, usability, and accessibility features of our fractal generation tool. Through optimizations in recursive processing, memory management, and user interface design, our tool offers clear advantages over traditional fractal generation programs like Fractint, particularly in terms of computational speed, ease of use, and hardware requirements.

Our fractal generation tool was developed with a focus on performance efficiency, making it capable of rendering complex fractal structures rapidly while using minimal system resources. The key efficiency features include:

1. We implemented caching mechanisms within the recursive algorithms to minimize redundant calculations. This approach reduces computation times significantly, especially for high-recursion fractals, such as tree structures and nested polygons, making our tool up to 33% faster than Fractint in execution time for similarly complex tasks;

2. Our tool dynamically allocates and reallocates memory based on the recursion depth and resolution settings, optimizing memory usage. For high-resolution images (e.g., 4000 × 4000 pixels), our tool required only 150 MB of memory, compared to Fractint’s 250 MB, achieving a 40% reduction in memory usage. This efficient memory handling enables our tool to operate smoothly on standard PCs, including those with limited RAM, making it practical for users across various fields;

3. To utilize the multi-core processors available in most modern PCs, our tool uses parallel processing for computationally intensive tasks, such as rendering complex fractal structures. This multi-threaded approach enhances responsiveness and allows the tool to maintain performance consistency, even with large recursion depths or high-resolution outputs;

4. Our tool allows users to adjust the fractal parameters in real time, providing immediate visual feedback without requiring a full regeneration of the fractal. This feature improves workflow efficiency, enabling users to experiment with variations quickly, which is especially valuable for designers who rely on real-time iteration;

5. For industrial applications requiring high-resolution exports, such as in textile or digital design, our tool incorporates buffered output methods that streamline disk I/O processes. This optimization reduces delays when saving large images, ensuring smooth operation for users working with extensive designs;

6. For low dependence on high-performance VGA, unlike Fractint, which requires high-performance VGA hardware for optimal operation, our tool was engineered to run efficiently on standard PC configurations without specialized graphics requirements. By optimizing CPU processing over GPU reliance, our tool makes complex fractal generation accessible to users without high-end graphics cards. This design broadens the tool’s applicability, allowing both professionals and students to work with fractals on a wide range of hardware setups.

Our fractal generation tool offers a user-friendly graphical interface that streamlines the process of fractal design, in contrast to the command-line interface of Fractint, which can be complex for users unfamiliar with command-line operations. To quantify the benefits of this interface, we conducted a usability study with the following findings:

• Task completion time: users completed standard tasks (e.g., setting parameters, exporting images) 30% faster with our tool than with Fractint, reflecting the convenience of an intuitive interface;
• Error reduction: the graphical interface reduced configuration errors by 40% compared to Fractint’s command-based input, improving accuracy and ease of use;
• Learning curve: new users reported becoming proficient with our tool’s basic operations within 10 min, compared to 25 min for Fractint, highlighting our tool’s accessibility;
• User satisfaction: in surveys, users rated our interface at 4.5/5 for ease of use, in contrast to Fractint’s 3.2/5, indicating greater satisfaction with our design.

A significant advantage of our tool is its compatibility with standard PC hardware. By reducing dependence on high-performance VGA, we ensure that complex fractal visualization and high-resolution output are feasible without specialized graphics hardware. This makes the tool accessible to a broader audience, including those using consumer-grade devices. Fractint, by contrast, relies on high-performance VGA for effective rendering, which limits its accessibility as shown in

Table 2. Performance comparison.

Metric	Our Tool	Fractint	Improvement
Execution Time (Depth 10)	2.7 s	4.0 s	33% faster
Memory Usage (4000 × 4000 px)	150 MB	250 MB	40% reduction
User Interface	Graphical, intuitive	Command-line, complex	Improved usability
Task Completion Time	15 s	22 s	30% faster
Error Rate	40% lower	Higher	More accuracy
Hardware Requirement	Standard PC	High-performance VGA	Greater accessibility

.

Through extensive optimizations and a user-centered design, our tool addresses many of the practical limitations encountered in traditional fractal generation software like Fractint. With a reduced reliance on specialized hardware, efficient resource management, and an accessible interface, our tool enables broader applications for fractal generation in industrial design, academia, and digital art.

These features make it an effective and versatile tool for users seeking both functionality and accessibility in fractal visualization.

To evaluate the performance of our proposed method against established fractal image-coding approaches, we compared it with Barnsley’s original IFS method and Jacquin’s fractal block-coding technique (see

Table 3. Comparative analysis with state-of-the-art fractal image coding methods.

Metric	Our Method	Barnsley’s IFS	Jacquin’s Method
Compression Ratio	Adjustable, high	High	High
Encoding Time (seconds)	12.5	15.8	18.3
Decoding Time (seconds)	10.2	12.5	14.7
PSNR (dB)	35.2	33.7	34.1
SSIM	0.92	0.88	0.90

). This analysis focuses on key performance metrics, including compression ratio, encoding and decoding time, and image quality, as measured by peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM).

• For the compression ratio, our method achieved a compression ratio comparable to Jacquin’s method while offering better control over the trade-off between image detail and compression rate. The fractal structures we generated allowed for flexibility in scaling compression levels based on application needs;
• For encoding and decoding time, benchmark tests indicated that our method has a faster encoding time compared to Jacquin’s method due to optimized recursive algorithms and efficient memory management. Our decoding process also required less computation, maintaining real-time rendering capability for high-resolution images. We conducted t-tests, comparing the encoding and decoding times of our method against Barnsley’s IFS and Jacquin’s methods. The results indicate a statistically significant reduction in both encoding time (p < 0.01) and decoding time (p < 0.01) for our method;
• For image quality (PSNR and SSIM), we measured image quality using PSNR and SSIM to provide quantitative comparisons of visual fidelity. Our method demonstrated a PSNR improvement of approximately 1.5 dB over Barnsley’s IFS method and achieved SSIM scores, indicating a higher structural similarity at comparable compression ratios. These metrics suggest that our method effectively preserves image quality, especially in terms of edge sharpness and structural consistency. T-tests comparing PSNR and SSIM values showed that our method produces significantly higher PSNR scores (p < 0.05) and SSIM values (p < 0.05) than Barnsley’s IFS, indicating better image quality and structural similarity. The higher PSNR and SSIM values reflect an improved preservation of image detail and structure with our approach.

In benchmark tests, our method required 12.5 s for encoding and 10.2 s for decoding on high-resolution images, compared to 15.8 s for encoding and 12.5 s for decoding using Barnsley’s IFS and 18.3 s for encoding and 14.7 s for decoding with Jacquin’s method see

Table 4. Summary of statistical testing results.

Metric	Comparison	p-Value	Significance
Encoding Time	Our Method vs. Barnsley’s IFS	p < 0.01	Significant
	Our Method vs. Jacquin’s	p < 0.01	Significant
Decoding Time	Our Method vs. Barnsley’s IFS	p < 0.01	Significant
	Our Method vs. Jacquin’s	p < 0.01	Significant
PSNR	Our Method vs. Barnsley’s IFS	p < 0.01	Significant
SSIM	Our Method vs. Barnsley’s IFS	p < 0.01	Significant
Compression Ratio	Our Method vs. Jacquin’s	p = 0.12	Not Significant

. These improvements reflect the lower computational demands of our approach, which is made possible by optimized recursion and adaptive memory allocation.

These statistical analyses confirm that the performance improvements provided by our method in terms of time efficiency and image quality (PSNR and SSIM) are statistically significant compared to standard approaches. This supports the effectiveness of our model and algorithm in achieving enhanced fractal image coding performance.

These comparative results illustrate that our method offers a competitive compression ratio and superior efficiency in both encoding and decoding times while also maintaining high image quality. By optimizing recursive processing and adapting fractal structures for image coding, our method achieves efficient fractal representation and enhances the practical applicability of fractal coding in digital imaging.

Our analysis underscores the advantages of the proposed method, particularly in scenarios requiring a balance between compression efficiency and image fidelity. These improvements demonstrate our method’s applicability in modern digital image-processing tasks where high-quality compression is essential

4.1. Fractal Dimension Analysis and Self-Similarity Verification Subsection

In this section, we analyze the fractal dimensions of the new patterns generated in our study, comparing them with well-known fractal dimensions, such as Hausdorff and Box-counting dimensions, of traditional patterns. This analysis is essential for understanding the mathematical complexity and scaling behavior of our fractals, as well as for verifying their self-similarity properties. We calculated the fractal dimensions of our generated patterns using both the Hausdorff dimension and Box-counting dimension methods. The dimensions of our new fractals fell within expected ranges for self-similar structures, yet they displayed distinctive values that differentiate them from traditional fractals. For example, a tree-shaped fractal generated in our study has a Hausdorff dimension of 1.86, compared to the traditional Sierpinski triangle’s Hausdorff dimension of 1.58. These differences highlight the complexity and uniqueness of our fractals.

To contextualize the novelty of our fractals, we compared the dimensions of our patterns with classic fractals, such as the Koch snowflake, Cantor set, and Sierpinski triangle as shown in

Table 5. Dimension analysis.

Fractal Pattern	Our Pattern Dimension	Traditional Dimension (Hausdorff/Box-Counting)	Comparison
Tree-shaped Fractal	1.86 (Hausdorff)	Sierpinski Triangle: 1.58	Higher Complexity
Circular Fractal	1.92 (Box-counting)	Koch Snowflake: 1.26	Higher Detail
Star-shaped Fractal	1.67 (Hausdorff)	Cantor Set: 0.63	Greater Density

. Our patterns, while similar in self-similarity properties, exhibited unique scaling behaviors, as evidenced by their differing dimensions. These distinctions underscore the mathematical uniqueness of our generated patterns and their potential for new applications in design and digital art. To visually verify the self-similarity properties of our fractals, we generated zoomed-in views of specific patterns. These images reveal that smaller sections of our fractals replicate the overall structure, confirming the self-similarity inherent in the recursive algorithms we used. This characteristic was observed consistently across various scales, further supporting the fractal nature of our patterns.

The fractal dimensions of our patterns generally exceed those of classic fractals, reflecting their increased structural complexity and design-richness. The consistent self-similarity of our fractals, combined with their unique dimension values, indicates that our patterns contribute new structural characteristics to the field of fractal geometry.

By conducting a mathematical dimension analysis and visually verifying self-similarity, we establish the uniqueness of our fractal patterns and provide quantitative and qualitative evidence of their fractal properties. This analysis reinforces the value of our generated patterns as distinct mathematical and visual contributions to fractal theory.

4.2. Real-World Applications and Comparison of Recursive Algorithm Properties

To demonstrate the versatility and effectiveness of our fractal generation method, we present case studies in three domains, including medical imaging, textile design, and digital art, each showcasing the practical utility of our patterns (see

Table 6. Real-world applications.

Domain	Application	Performance Metrics	Results
Textile Design	Fabric Pattern Creation	Aesthetic quality, scalability, production feasibility	Higher aesthetic score (4.7/5), seamless scalability
Digital Art and Architecture	Tile and Facade Design	Visual appeal, customization, structural integrity	Increased visual appeal (5/5), adaptable across scales

). For each application, we define domain-specific performance metrics to highlight the advantages of our approach over traditional fractal generation techniques. Fractal patterns provide unique, scalable designs in the textile industry, particularly for fabric printing and pattern creation. We applied our method to generate complex, culturally inspired textile patterns and evaluated them using aesthetic quality, scalability, and production feasibility.

Using our tool, we created a series of textile designs inspired by traditional motifs, generating high-resolution images with intricate, self-similar patterns. Designers reported a higher aesthetic quality (scored 4.7/5) and an increased scalability of our designs compared to traditional patterns. The production process was also simplified due to the seamless scalability of our fractals, allowing for efficient pattern adaptation across different fabric sizes and types. In digital art and architecture, fractal structures provide visually engaging and structurally complex designs. We applied our method to create architectural motifs and digital art patterns, evaluating the results based on visual appeal, customization ease, and structural integrity. Architectural patterns generated by our tool were used to create designs for ceramic tiles and facade elements. Our fractal designs, when compared to standard geometric patterns, demonstrated higher visual appeal (5/5 in user surveys) and provided customization options that allowed architects to scale and adapt the patterns across various surface areas. Digital artists using our tool also appreciated the high level of customization, which allowed them to create dynamic, multi-layered visual effects in a fraction of the time needed for manual pattern creation.

These case studies demonstrate the adaptability of our fractal generation method to various domains, with significant benefits over traditional techniques. The unique structural complexity and self-similarity properties of our fractal patterns make them especially suitable for applications requiring detailed, scalable, and visually appealing designs. By defining and meeting application-specific metrics, we show that our method can serve as an effective tool for practical use in multiple industries. Moreover, to substantiate the novelty of our recursive algorithms, we analyzed their convergence properties, symmetry, and precision relative to conventional fractal-generating algorithms. Our approach combines optimized recursion and adaptive memory handling, resulting in faster convergence and higher precision under specific conditions, as demonstrated through complexity theory analysis.

Traditional fractal generation algorithms, such as Barnsley’s IFS, often require a high number of iterations to achieve convergence, with the time complexity typically around O(n²). By contrast, our algorithms integrate caching and in-place calculations that reduce redundancy, resulting in a convergence complexity of O(n). The experimental results show that our method converges approximately 30% faster for high-resolution fractals with deep recursion levels (see

Table 7. Units for magnetic properties.

Property	Our Recursive Algorithm	Conventional Algorithms	Improvement
Convergence Complexity	O(n)	O(n2)	30% faster convergence
Symmetry Deviation	2%	5%	Improved structural balance
Precision (Dimensional Deviation)	<1%	~3%	Higher precision in structural detail
Iteration Complexity	O(nlogn)	O(n2)	Reduced iteration count

). These properties make our approach particularly effective for applications requiring rapid convergence, such as real-time fractal visualization. Conventional recursive methods often face challenges in maintaining symmetry across iterations, especially in fractals with non-standard or culturally inspired designs. Our algorithms incorporate controlled scaling and transformation adjustments to maintain symmetry across each recursion depth. By tracking transformation matrices throughout recursion, our method ensures symmetrical balance in complex structures, such as star- and tree-shaped fractals. The symmetry of our fractals was quantitatively evaluated by measuring deviations from the central axis, with our method achieving an average deviation of 2%, compared to the 5% in conventional methods, reflecting a higher degree of symmetry.

We assessed the precision of our algorithm by calculating the fractal dimension and comparing it to theoretical values. The adaptive nature of our algorithm improves precision, especially for fine-detail fractals like circular and intersecting patterns. This improved precision was validated by comparing calculated dimensions, where our method’s deviation from expected theoretical values was less than 1%, compared to the 3% in standard algorithms.

Utilizing tools from complexity theory, we evaluated the performance of our algorithms based on the number of iterations required for convergence and the associated computation cost. Our recursive approach, with its reduced iteration requirement and in-place calculations, achieves lower computational complexity. For example, while conventional methods typically require O(n²) iterations for detailed fractals, our approach completes similar iterations with O(nlogn) complexity, enabling faster generation of high-detailed fractals with fewer computational resources.

These improvements make our recursive algorithms more suitable for applications requiring efficient, symmetrical, and precise fractal generation. By reducing the convergence time, ensuring symmetrical alignment, and enhancing structural precision, our method provides a distinct advantage over traditional approaches, especially for real-time applications in design, architecture, and scientific visualization.

5. Conclusions

The study of fractals through mathematical modeling and recursive algorithms reveals their profound potential in design and industry. By expanding the discussion to include global patterns and applications, this research underscores the universal relevance of fractal geometry. The ability to generate intricate patterns through simple iterative processes allows for the creation of aesthetically pleasing and functionally effective designs. This research highlights the significance of fractal construction methods, their visualization algorithms, and their practical applications, demonstrating the intersection of mathematics and art in modern and traditional design practices worldwide.

The following main conclusions were presented regarding the results of the work:

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The basic concepts of the theory of fractals, the origin of fractals, their properties, types, fields of application, and methods of calculating the fractal dimension were researched. As a result, it was found that national patterns with a fractal structure can be used in the organization of design processes in modern and classic design in light industry and have economic benefits;

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Fractal construction methods, their visualization algorithms, the application process, the concept of fractal measurement, and methods of calculating the size of objects with a fractal structure were investigated. Based on iterative methods and the iterative function system (IFS), the theoretical basis and advantages of the L-systems method were used to develop geometric models of complex fractals. It made it possible to perform a comparative analysis of classical and modern fractals built using these methods;

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A research model of objects with a fractal structure was built using the RFM method, and an algorithm was developed. Based on this algorithm, circular, tree-shaped, star-shaped, and polygonal fractals from classic fractals were visualized. Fractal dimensions of fractal shapes were determined using the developed models. This method and model made it possible to create a wide range of models of complex geometric shapes;

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Based on the developed algorithms, a software tool for automating the visualization of objects with a fractal structure was developed. This software tool made it possible to automate the creation of patterns of gauze and carpets in the textile industry, interior and exterior decoration, furnishing and design of rooms in the construction of buildings, the decoration of construction products with patterns, their design, and the selection of colors and patterns for their products.