FRACTAL STRUCTURED GEOMETRIC PATTERNS: DESCRIPTION AND CONSTRUCTION METHODS

ABSTRACT

This article conducts a comparative analysis of complex-configured fractal patterns, their types, construction methods, applications in interior and architectural design, and the relationship between fractal geometry and design, presenting several results. The relevance of the article lies in the fact that the use of fractal geometry in modern interior design is considered a new and promising direction. Fractal patterns help to integrate complexity, consistency, and natural aesthetics, positively influence human psychology, reduce stress, and create a comfortable environment in the space. We proposed creating patterns using the method of geometric shapes and geometric transformations, achieving several results. The article explores fractal geometry, complex structured patterns, fractals, geometric shapes and transformations, the construction of fractal patterns through recursive functions and analytical formulas, and their applications in the fields of interior design and architecture, with conclusions drawn.

АННОТАЦИЯ

Данная статья проводит сравнительный анализ фрактальных узоров сложной конфигурации, их типов, методов построения, применения во внутреннем и архитектурном дизайне, а также связи фрактальной геометрии с дизайном, представляя несколько результатов. Актуальность статьи заключается в том, что использование фрактальной геометрии в современном интерьере является новым и перспективным направлением. Фрактальные узоры помогают объединить сложность, последовательность и естественную эстетику, оказывая положительное влияние на психологию человека, снижая уровень стресса и создавая комфортную атмосферу в помещении. Мы предложили создавать узоры с использованием метода геометрических форм и геометрических преобразований и получили ряд результатов. В статье рассмотрены фрактальная геометрия, узоры сложной структуры, фракталы, геометрические формы и преобразования, построение фрактальных узоров с использованием рекурсивных функций и аналитических формул, а также их применение в области интерьера и дизайна. Сделаны выводы.

Keywords: fractal geometry, algorithms, dynamic design, Euclidean geometry, geometric patterns, automation, sustainability, geometric transformations, squeezing coefficient

Ключевые слова: фрактальная геометрия, алгоритмы, динамический дизайн, евклидова геометрия, геометрические узоры, автоматизация, устойчивость, геометрические преобразования, коэффициент сжатия

Introduction

Fractal geometry plays an important role in describing natural systems, especially in understanding dynamic relationships within complex systems. By studying complex systems associated with natural patterns and forms, fractal geometry holds significant value in interior and architectural design theories. This article explores the application of fractal geometry in interior and architectural design and analyzes practical applications related to it. The aim of the research is to examine how fractal geometry design, utilizing complex patterns, can be integrated into contemporary interior design. Fractal geometry is one of the best methods for representing natural forms, allowing for the creation of complex shapes through rules and steps. Although fractal geometry is a branch of mathematics, today it serves as a source of inspiration in interior design, furniture design, and architectural design, thanks to its aesthetic appeal and innovative approach to depicting natural objects.

Fractal geometry is a branch of mathematical geometry that studies complex shapes and patterns. Fractals often exhibit a property of self-similarity, where each part of a fractal, when magnified, replicates the original shape. Fractal geometry is used to describe shapes found in nature, such as mountains, clouds, trees, and river systems, as these forms also have complex structures and often display self-similarity despite variations in scale.

Benoit Mandelbrot introduced the term "fractal" and described it as follows: “A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is, at least approximately, a reduced-size copy of the whole.”[1]

In the literature [2], information on fractal theory and its areas of application is compiled. It emphasizes the importance of understanding fractal theory, as fractal models emerge in nonlinear processes and the processes described by fractals are controlled in practical applications.

In architecture, fractals can be used to model the structure of buildings and to create decorative patterns on walls. This process can involve shapes from Euclidean geometry or fractal patterns found in nature. The use of fractals in architecture can reduce the demand for designers in this field, allowing for the rapid development of new, modern structures [3].

The article [4] provides instructions, examples, and algorithms for methods of creating fractals in fractal geometry, featuring stunning fractal images. Specifically, it demonstrates methods for creating fractals such as the Mandelbrot set, tree-like fractals, heart-shaped fractals, the Julia set, and tall towers based on the Mandelbrot set. The images are developed with consideration of certain operations, such as geometric iteration rules and sequential removal. In addition to generation functions, particular attention is given to examining their corresponding initial conditions and the number of iterations. The program created was tested for all cases. Mathematica and MATLAB were used to develop the program.

While Euclidean geometry describes lines, ellipses, circles, and so on using equations, fractal geometry represents objects from the perspective of recursive algorithms. One method for constructing fractals is through Iterated Function Systems, or IFS. IFS follows a general approach of transforming a geometric object in a specific way, leaving behind several smaller objects, each similar to the original. The process is then repeated on each of these smaller objects to create even smaller parts, and so on. A fractal is the result of performing this process infinitely many times [5].

Currently, there are two main methods for constructing fractal equations: one involves the use of mathematical methods and computer programs, and the other is through fractal design software, such as Apophysis, Ultra Fractal, Fractal Ferry Man, and others [6].

Methods

We proposed creating patterns using the method of geometric shapes and geometric transformations. There are three main types of iteration in generating and modifying fractals:

1. Generator iteration – repeatedly generating the fractal by replacing certain geometric shapes with other shapes.

2. IFS iteration – developing the fractal by repeatedly applying transformations (such as rotation and reflection) to points.

Before the introduction of the concept of fractal geometry, geometric models of natural objects were represented using combinations of simple shapes such as straight lines, triangles, circles, rectangles, and spheres. However, it is challenging to describe more complex natural objects, such as porous materials, cloud shapes, and tree forms, using Euclidean geometry. Today, the study of the mathematical aspects of fractal theory, as well as methods for describing natural processes and phenomena using fractal theory, has become an independent field of science. When studying fractals, it is difficult to draw a clear boundary between mathematics and computer science, as they are closely interconnected, both aiming to discover unique, non-repetitive patterns. The study of fractals enables a better understanding of certain natural processes and phenomena. The application of fractal shapes in modern computer graphics is considered essential. By adjusting parameters in given equations, it is possible to create highly complex fractal shapes.

Let’s assume that the basis of complex-configured fractals consists of geometric patterns. The repetitions in these complex [7] patterns can also be divided into several types, as shown in Figure 1.

Figure 1. Geometric patterns and their repetition methods

Results and discussion

In developing complex patterns based on fractals, we create intricate structures by repetitively drawing geometric shapes and achieved several results.  These patterns are composed of identical shapes, referred to as Unit Forms. These shapes are repeated at each step in the design process (with n- representing the number of steps), adding a unique aesthetic quality to the design. The use of Unit Forms in design helps to create an overall shape or pattern using identical elements (Figure 2). This process allows designers to unify the overall composition.

Figure 2. Creating a fractal-patterned design using geometric shapes

Repetition of Unit Forms: Design is typically based on the repetition of identical units, which establishes rhythm and harmony. The repetition of identical unit shapes makes the design more complex and aesthetic. A high frequency of unit repetitions helps the design achieve perfect harmony and form (Figure 3). These patterns are created through the repetition of geometric elements and enable the formation of intricate fractals in interior design.

Figure 3. Creating a fractal-patterned design using geometric shapes

The formulas and concepts provided below help to mathematically represent the process of creating complex-structured fractal patterns and are expressed as follows:

-center point

- base radius

 – count edges

 - starting angle

- step

 - squeezing coefficient  

Based on this formula, we obtain various forms of fractal patterns depending on the values of the parameters. (Figure 4)

Figure 4. Variations of a fractal with an initial shape as a triangle at different values

Conclusions

In conclusion, mathematical approaches and computer technologies play a significant role in the creation of fractals. The process of constructing fractal patterns involves recursive algorithms, where a shape is generated by starting with an initial geometric element and then repeating it at successive scales. Due to their rule-based nature and potential for infinite detailing, fractals are widely used in art, design, and architecture. Fractal patterns exhibit complexity and consistency in their appearance. Understanding these patterns and learning the methods of constructing them further expands the possibilities of applying them in modern graphic technologies and visual representations. Thus, fractal-structured geometric patterns are not only aesthetically intriguing but also hold great significance for scientific research and technological advancements. The methods for creating these patterns contribute to their broader application and support the development of new innovative solutions.

References:

1. Mandelbrot B.B. Fraktal'naya geometriya prirody. - M.: Institut komp'yuternykh issledovaniy, 2002. S – 656. [in Russian].
2. Liu, Shu Tang, Yong Ping Zhang, and Chang An Liu. Fractal control and its applications. Springer Nature, 2020.
3. Alik B., Ayyildiz S. Fractals and Fractal Design in Architecture //13th international conference “standardization, prototypes and quality: A means of Balkan countries’ collaboration”. November. – 2016. – С. 3-4.
4. Ahammad Hossain, Nurujjaman, Rita Akter, and Payer Ahmed. "Fractals Generating Techniques." Sonargaon University Journal Volume 1, Number 2, December 2016.43-54 p.
5. Jianzhang Zhao, Jing Shen. Application of Fractal Geometry in Textile Digital Printing Pattern Design. 2021 Fifth International Conference on I-SMAC (IoT in Social, Mobile, Analytics and Cloud) (I-SMAC). Palladam, India November 2021, DOI: 10.1109/I-SMAC52330.2021.9640699
6. Anarova, Sh.A., Ibrohimova, Z.E. "The Use of Software Tools for Creating Fractal Patterns in Textile Design." Collection of Reports from the National Scientific-Technical Conference “Innovative Ideas in Information and Communication Technologies and Software Development,” Samarkand – 2020. – pp. 10-14.
7. Eman Sarkan, Design methods in the application of fractal geometry in the interior design of tourist buildings. // International Design Journal, Volume 7, Issue 3, July 2017. 92-93 pp.