Formation of Periodic Mosaic Structures Using Operations in Galois Fields

Mosaic ornaments and periodic geometric patterns are deeply rooted in cultural heritage and contemporary design, where symmetry plays a fundamental role in both aesthetic and cognitive perception. This study develops an algebraic method for generating symmetrical and periodic mosaic structures using operations in Galois fields. The approach demonstrates that the intrinsic properties of finite fields naturally give rise to symmetry and periodicity, eliminating the need for specific initial patterns, even when applied to relatively simple algebraic expressions such as the Bernoulli lemniscate and the cissoid of Diocles. The proposed algorithm offers the advantages of simplicity and the ability to provide gradual transitions from one mosaic structure to another. Furthermore, it is demonstrated that standardization of algebraic expressions used for mosaic generation can be efficiently achieved through discrete logarithm operations. A novel method for computing discrete logarithms is introduced. The results confirm that symmetrical structures of high complexity can be obtained through simple expressions, and their periodicity becomes more pronounced with increasing field characteristics. This approach offers practical applications in textile and wallpaper design, smart materials, and psychological testing, while also suggesting new perspectives for the analysis of mosaic-like natural systems where symmetry is a defining property.