DEVELOPMENT OF JEWELRY WITH FRACTAL STRUCTURES

ABSTRACT

This article covers the process of designing fractal necklaces and bracelets using geometric shapes and geometric substitution. The combination of fractal geometry and geometric transformation methods opens up new aesthetic and functional possibilities in jewelry design. This approach provides the basis for innovative designs in jewelry design, creating necklaces and bracelets with aesthetically interesting and complex fractal structures. It also covers the basic mathematical principles used to create fractal patterns, including the use of parameters such as radius, shrinkage factor, repetition step, and angle. Simple shapes are transformed into complex fractal structures using geometric transformations such as scaling, rotation, and translation operations. This approach creates self-repeating patterns that make necklace and bracelet designs more attractive and unique. Overall, this article explores the creative possibilities associated with the use of fractal geometry in jewelry art and design, and shows how this approach can add complexity, beauty, and originality to jewelry.

АННОТАЦИЯ

Эта статья посвящена процессу разработки фрактальных цепочек и браслетов с использованием метода геометрических форм и геометрических преобразований. Комбинация фрактальной геометрии и метода геометрических трансформаций открывает новые эстетические и функциональные возможности в дизайне ювелирных изделий. Такой подход способствует созданию инновационных дизайнов ювелирных украшений с эстетически привлекательными и сложными фрактальными структурами цепочек и браслетов. В статье также освещены основные математические принципы, используемые при создании фрактальных узоров, включая параметры, такие как радиусы, коэффициенты сжатия, итерационные этапы и углы. Посредством геометрических преобразований, таких как масштабирование, вращение и перенос, простые формы преобразуются в сложные фрактальные структуры. Этот подход позволяет создавать самоповторяющиеся узоры, которые делают дизайн ожерелий и браслетов более привлекательным и уникальным. В целом, статья исследует творческие возможности применения фрактальной геометрии в ювелирном искусстве и дизайне, а также демонстрирует способы добавления сложности, красоты и уникальности ювелирным изделиям через этот подход.

Keywords: Fractals, jewelry, fractal geometry, geometric transformations, fractal necklace, fractal bracelet, fractal design.

Ключевые слова: Фракталы, украшения, фрактальная геометрия, геометрические перестановки, фрактальная цепочка, фрактальный браслет, фрактальный дизайн.

INTRODUCTION

Fractals surround various aspects of our lives. As the term has become more widely used, interest in understanding what fractals are and how they affect our lives is growing every day. The discovery of fractals marks the opening of a new aesthetic in science, mathematics, art, as well as a breakthrough in human perception of the world [1, 3]. Today, the study of the mathematical aspects of fractals, as well as the use of fractal theory to describe natural processes and phenomena, has evolved into an independent field of modern science. The theory of fractals has expanded so much that it is now studied across several specialized fields. Specifically, fractal geometry is successfully applied in areas such as astronomy, chemistry, biology, economics, medicine, computer graphics, telecommunications, radio engineering, architecture, and jewelry design [4]. Jewelry has always attracted attention worldwide with its beauty, elegance, and sophistication. It intricately connects elements of decorative arts, artistic creativity, craftsmanship, and modern manufacturing. Creating jewelry based on fractals represents a prominent genre of design and art. Fractal artworks are often developed by using methods such as iterative digital processes and random parameter selection, offering designers limitless artistic creativity and unique patterns. In many cases, these patterns are developed using fractal-generating software, and the process consists of three stages: setting parameters for the desired fractal, performing calculations, and obtaining the results [2].

METHODS AND REULTS

The use of fractal geometry in jewelry design is not merely an expression of mathematical beauty but rather a study of the harmony between form and function. The self-similar patterns inherent to fractals can be used to create visually appealing and structurally robust designs. By progressively replacing simple forms with increasingly complex patterns, we have developed intricate fractal shapes suitable for fractal necklaces, applying fundamental concepts of geometry (Figure 1). The formulas and concepts provided below help mathematically represent the creation process of a fractal necklace, and are expressed as follows: 

L – Length of the fractal necklace,  - contraction coefficient of the left part of the fractal necklace,  - contraction coefficient of the bottom part of the fractal necklace, - contraction coefficient of the right part of the fractal necklace,  - steps of the left part of the fractal necklace,  - steps of the right part of the fractal necklace,  - steps of the bottom part of the fractal necklace.

As a result, we obtain the fractal necklace shown in Figure 1 at various steps of the iteration.

In developing fractal-based bracelets, we also considered creating complex structures using the method of geometric transformations. The method of geometric transformations typically involves using various geometric processes in constructing fractals. The primary operations include: scaling - to change the size of shapes; rotation - to rotate the shape by an angle; translation - to change the position of the shape. In this method, the applied geometric shape or figure is drawn in a repetitive and recursive manner. At each step, geometric transformations are applied, resulting in sequentially transformed shapes that form a fractal. In this approach, the process depends on predetermined geometric transformations based on specific rules [5]. The formulas and concepts provided below help mathematically represent the creation process of a fractal bracelet and they are expressed as follows:

 - Radius of the fractal bracelet’s circles. , - positions (positions of the functions  and );  - number of steps;  - number of angles to be drawn;

Based on this formula, we obtain various designs of fractal bracelets (Figure 2).

These mathematical formulas are used to create fractal bracelets through the method of geometric transformations. Shapes are developed at each step based on radiuses and coefficients, and they are iteratively assembled, resulting in the image of a fractal bracelet.

DISCUSSION Jewelry pieces based on fractal geometry, including necklaces and bracelets, occupy a unique place in modern design. This approach enables the creation of complex, self-repeating patterns by using simple geometric shapes. The geometric transformation method produces intricate fractal patterns from simple shapes, where forms are reduced or rotated over several stages to develop unique designs for necklaces and bracelets. Fractal design serves not only decorative purposes but also as a means to explore mathematically intriguing shapes. Additionally, such jewelry items convey a sense of completeness, offering extensive opportunities for contemporary and innovative designs. Overall, developing necklaces and bracelets through geometric transformation methods and fractal structures opens new creative doors in the field of jewelry, with research in this area contributing to the enrichment of jewelry art.

CONCLUSION

The application of fractal geometry in jewelry design, particularly for necklaces and bracelets, represents a significant innovation in modern aesthetics and functional artistry. Through the use of geometric transformation methods, such as scaling, rotation, and translation, complex fractal patterns are developed from simple geometric shapes. These patterns offer not only a visually appealing design but also structural integrity and uniqueness. By iteratively applying mathematical formulas, shapes are progressively refined, resulting in intricate fractal necklaces and bracelets that merge art, mathematics, and technology. This approach demonstrates how fractal geometry can transcend its theoretical roots to become a tool for creativity and innovation in jewelry craftsmanship. Ultimately, fractal-based jewelry design contributes to the enrichment of the field, providing limitless opportunities for creating distinctive and aesthetically engaging pieces. The research and application of these methods open new possibilities for designers, merging traditional artistry with cutting-edge computational techniques.

References:

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